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This article is cited in 1 scientific paper (total in 1 paper) Canonical form of the M. I. Belishev, A. V. Kaplun St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2022, Volume 86, Issue 4, DOI: https://doi.org/10.4213/im9179 
Bibliographic databases:
    § 1. Introduction1.1. On this paperThere is an approach, called the boundary control method (BC-method) [1], to inverse problems of mathematical physics. The approach has a pronounced interdisciplinary character: it is based on the connections of inverse problems with system theory and control theory and uses asymptotic methods, functional analysis, operator theory, etc. An algebraic version of the BC-method, based on connections with Banach algebras, provided a new solution to the problem of recovering a Riemannian manifold from boundary data [1]–[3]. Our perspective goal is to apply this version to inverse problems on graphs. The present paper is a step in this direction. The algebraic version is based on the following fundamental fact: a topological space can be characterized in terms of an appropriate algebra. For example, a compact Hausdorff space $\Omega$ is determined (up to homeomorphism) by the algebra $\mathfrak{A}=C(\Omega)$ of continuous functions (Gel’fand, 1943). Moreover, the spectrum of this algebra, i.e., the set $\widehat{\mathfrak{A}}$ of its irreducible representations endowed with a suitable topology, is homeomorphic to the original space: $\widehat{\mathfrak{A}}\cong\Omega$. As a consequence, by taking any copy $\mathfrak{A}'$ of the algebra $\mathfrak{A}$ and finding its spectrum $\widehat{\mathfrak{A}'}\cong\widehat{\mathfrak{A}}\cong\Omega$, we obtain a homeomorphic copy of the space $\Omega$. This enables us to solve the reconstruction problem: a copy $\mathfrak{A}'$ is extracted from the inverse problem data, and its spectrum $\widehat{\mathfrak{A}'}$ provides the solution to the reconstruction problem: a homeomorphic copy of the manifold $\Omega$ is recovered. An important part of the approach consists in finding the algebra $\mathfrak{A}'$ in terms of the known data. For this algebra, we take the eikonal algebra $\mathfrak{E}$ determined by the dynamical system which describes the wave propagation in $\Omega$. Variants of the BC-method for inverse problems on graphs are proposed in [4]–[6]. The version using the eikonal algebra was suggested in [7] and extended in [8]. Our work develops this approach. Its general direction is to study the relations between the properties of the algebra $\mathfrak{E}$ (block structure, algebraic invariants, representations) and the graph geometry. A promising goal is the recovery of a graph from its boundary data. 1.2. The eikonal algebraFor clarity and without loss of generality, we can regard $\Omega$ as a connected compact graph in $\mathbb{R}^3$ consisting of smooth curves (edges) $\{e_1,\dots,e_l\}=E$, which are joined at interior vertices1 $\{v_1,\dots,v_m\}=V$. There are boundary vertices $\{\gamma_1,\dots,\gamma_n\}=\Gamma$, with a single edge outgoing from each of them. The metric (inner distance) in $\Omega$ is induced by the Euclidean metric of $\mathbb{R}^3$. The edges of the graph are “material”: oscillations (waves) propagate along them, being initiated by point sources (controls), which are placed at the boundary vertices. The waves move from the boundary with unit velocity, gradually filling the graph. The process is described by the dynamical system
$$
\begin{equation*}
\begin{alignedat}{2} &u_{tt}-\Delta u=0 &\quad &\text{in }\mathscr{H},\quad 0<t<T, \\ &u|_{t=0}=u_t|_{t=0}=0 &\quad &\text{in }\Omega, \\ &u=f &\quad &\text{on }\Gamma \times [0,T], \end{alignedat}
\end{equation*}
\notag
$$
where $\mathscr{H}=L_2(\Omega)$, $\Delta$ is the Laplacian defined on smooth functions that satisfy the matching (Kirchhoff) conditions at the interior vertices; $f=f(\gamma,t)$ is a boundary control of the class $L_2(\Gamma\times [0,T])=: \mathscr{F}^T$; $u=u^f(x,t)$ is a solution (wave), $u^f(\,{\cdot}\,,t)\in\mathscr{H}$ at $0\leqslant t\leqslant T$. It is possible to control waves coming from a part $\Sigma\subset\Gamma$ instead of the whole boundary. In this case, we use controls of the class
$$
\begin{equation*}
\mathscr{F}^T_\Sigma:=\{f\in\mathscr{F}^T\mid \operatorname{supp} f\subset \Sigma\times[0,T]\}=\bigoplus\sum_{\gamma\in\Sigma}\mathscr{F}^T_\gamma.
\end{equation*}
\notag
$$
For every boundary vertex we have the family of reachable sets $\mathscr{U}^t_\gamma: =\{u^f(\,{\cdot}\,,t)\mid f\in\mathscr{F}^T_\gamma\}$, $0\leqslant t\leqslant T$, and the corresponding projectors $P^t_\gamma$ in $\mathscr{H}$ to $\mathscr{U}^t_{\gamma}$. The operator $E^T_\gamma:=\int_0^Tt\,dP^t_\gamma$ is called the eikonal corresponding to the vertex $\gamma$. The eikonals are self-adjoint operators belonging to the algebra $\mathfrak{B}(\mathscr{H})$ of bounded operators. Given a $C^*$-algebra $\mathfrak{A}$ and a set $S\subset\mathfrak{A}$, we write $\vee S$ for the $C^*$-algebra generated by this set, i.e., the minimal $C^*$-subalgebra in $\mathfrak{A}$ containing $S$. The eikonal algebra corresponding to a family $\Sigma\subset\Gamma$ of boundary vertices is the operator $C^*$-algebra
$$
\begin{equation}
\mathfrak{E}^T_\Sigma:=\vee\{E^T_\gamma\mid\gamma\in\Sigma\}\subset\mathfrak{B}(\mathscr{H}).
\end{equation}
\tag{1.1}
$$
1.3. Results and commentsIt was established in [7] that the algebra $\mathfrak{E}^T_\Sigma$ has a block structure: it is isomorphic to some subalgebra of $\bigoplus_{j=1}^J C([0,\epsilon_j];\mathbb M^{m_j})$ and differs from the latter by the existence of relations between blocks. For the simplest graphs (three-beam stars). the nature of these relations and their evolution with respect to $T$ were considered in [8]. The main result of this paper is a canonical block form of the eikonal algebra. This form is, firstly, distinguished by the absence of relations between blocks and, secondly, invariant: up to trivial transformations (permutation of blocks, change of parameterization, etc.), it is determined by any copy of the algebra (1.1). This enables us to hope that $\mathfrak{E}^T_\Sigma$ may be useful in inverse problems. whose data determine it up to isometry. Of course, the effectiveness of the approach can be fully judged only by seeing concrete applications to inverse problems. There are no such applications in this paper but the above results seem to be an important step in this direction. The eikonal algebra belongs to the class of $C^*$-algebras with finite-dimensional representations of different dimensions [9], [10]. The reduction to a canonical form is equivalent to constructing a functional model which realizes $\mathfrak{E}^T_\Sigma$ as an algebra of continuous matrix-valued functions on its spectrum. It belongs to the class of models described by Vasil’ev [9]. The definition (1.1) reproduces the definition of the corresponding algebras used in [1]–[3] for reconstruction of manifolds. The success of this application motivated our attempt to transfer this approach to problems on graphs. The non-commutativity of the algebra $\mathfrak{E}^T_\Sigma$ is an obstacle. It has already been encountered in the inverse problem of electrodynamics [3], but there it was solved by taking a quotient by the ideal of compact operators and thus reducing the problem to the commutative algebra $C(\Omega; \mathbb R)$. The non-commutativity of $\mathfrak{E}^T_\Sigma$ is irremovable, which makes its study much more difficult. Inverse problems on graphs are a topical subject. Various formulations and approaches were given by Avdonin, Kurasov, Novachik, A. Mikhailov and V. Mikhailov, Kuchment, Yurko. The BC-method and other approaches were used in [11]–[17] to solve dynamic and spectral inverse problems for various classes of graphs. Yurko and his followers used a spectral approach to inverse problems for differential operators on graphs [18]–[20]. We mention an informative review [21] by Kuchment and Berkolayko on the whole subject of quantum graphs, including inverse problems on them. The introductory part of the present paper is rather long and essentially repeats the corresponding sections of [7] and [8]. This is unavoidable since the presentation of facts and results related to $\mathfrak{E}^T_\Sigma$ requires a solid preparation. The technical part is rather complicated since we deal with an extremely general object: an arbitrary compact connected metric graph having a boundary. The study of simple examples in [7], [8] with illustrations may be instructive. We also recommend [22], where the reduction of $\mathfrak{E}^T_\Sigma$ to a canonical form is described in detail for a simple graph. This paper is addressed to specialists in $C^*$-algebras with a taste for applications and/or specialists in the field of mathematical physics who share the idea of the usefulness of abstractions. The eikonal algebra is a complex and many-faceted object worthy of a thorough study. The possible relations between its algebraic invariants and graph geometry are particularly interesting in inverse problems. For example, we mention a conjecture on the correspondence between clusters in the spectrum of $\mathfrak{E}^T_\Sigma$ and interior vertices of $\Omega$. It is very interesting how the presence of cycles in the graph affects the structure of the spectrum. § 2. Waves on graphs2.1. GraphsLet $\Omega=E\cup W$ be a connected compact graph in $\mathbb{R}^3$ with edges $\{e_1,\dots,e_L\}=E$ and vertices $\{w_1,\dots,w_M\}=W$. The edges are smooth2 curves whose ends are vertices. It is convenient to think of edges as open without including their ends. A vertex $w$ and an edge $e$ are incident (we write $w\prec e$) if $w$ is an end of $e$. The vertices $\{\gamma_1,\dots,\gamma_N\}=\Gamma$, each of which is incident to a single edge, are boundary vertices, and those in the set $\{v_1,\dots,v_{M-N}\}=V=W\setminus\Gamma$ are interior. The valence of a vertex $w$ is the number $\mu(w)$ of edges that are incident to $w$. When $\gamma\in \Gamma$, we have $\mu(\gamma)=1$. We also assume that there are no vertices with $\mu(w)=2$. Hence, $\mu(v)\geqslant 3$ for all interior vertices. The graph is equipped with the metric (internal distance) $\tau$ induced by the Euclidean metric of $\mathbb{R}^3$. Thus $\tau(a,b)$ is the minimum length of piecewise smooth curves lying in $\Omega$ and connecting $a$ and $b$. Given a set $A\subset\Omega$, we denote its metric neighborhood of radius $r$ by
$$
\begin{equation*}
\Omega^r_A:=\{x\in\Omega\mid \tau(x,A)<r\},\qquad r>0.
\end{equation*}
\notag
$$
Each edge $e$ is parameterized by the arclength $\tau$ beginning at one of its ends. For a function $y$ on the graph, the sign of its derivative $dy/d\tau$ with respect to arclength depends on the end chosen, but the second derivative $d^2y/d\tau^2$ is independent of this choice. Given a vertex $w$ and an incident edge $e$, we define the derivative in the direction outgoing from $w$,
$$
\begin{equation*}
\biggl[\frac{dy}{d\tau}\biggr]^+_e(w):=\lim_{e\ni x\to w}\frac{dy}{d\tau}(x).
\end{equation*}
\notag
$$
The quantity
$$
\begin{equation*}
F_w[y]:=\sum_{e\succ w}\biggl[\frac{dy}{d\tau}\biggr]^+_e(w)
\end{equation*}
\notag
$$
is called the flow of function $y$ through vertex $w$. The metric $\tau$ on a graph determines a (real) Hilbert space $\mathscr{ H}=L_2(\Omega)$ with scalar product
$$
\begin{equation*}
(y,u)_\mathscr{H}:=\int _\Omega y u\,d\tau = \sum_{e \in E} \int _e y u\,d\tau.
\end{equation*}
\notag
$$
Let $C(\Omega)$ be the space of continuous functions with norm $\|y\||\,{=}\sup_\Omega|y(\cdot)|$. A function $y$ belongs to the Sobolev class $H^2(\Omega)$ if $y\in C(\Omega)$ and $dy/d\tau,d^2y/d\tau^2\in L_2(e)$ on each edge. Define the Kirchhoff class
$$
\begin{equation*}
\mathscr{K} := \{y \in {H}^2(\Omega)\mid F_v[y]=0,\, v\in V \}.
\end{equation*}
\notag
$$
The Laplace operator on a graph is given by
$$
\begin{equation}
\Delta\colon \mathscr{H}\to\mathscr{H}, \quad \operatorname{Dom}\Delta=\mathscr{K},\qquad (\Delta y)|_e=\frac{d^2y}{d\tau^2}, \quad e \in E.
\end{equation}
\tag{2.1}
$$
It is densely defined and closed. 2.2. WavesThe initial-boundary value problem describing the wave propagation on a graph is of the form
$$
\begin{equation}
u_{tt}-\Delta u=0 \quad \text{in }\mathscr{H},\quad 0<t<T,
\end{equation}
\tag{2.2}
$$
$$
\begin{equation}
u(\,{\cdot}\,,t)\in \mathscr{K} \quad \text{for }0\leqslant t\leqslant T,
\end{equation}
\tag{2.3}
$$
Here $T >0$ is the final moment of time, $f=f(\gamma,t)$ is the boundary control and $u=u^f(x,t)$ is a solution (wave). If the control $f$ is smooth in $t$ and vanishes near $t=0$, the problem has a unique classical solution $u^f$. By the definition (2.1), the solution $u^f$ satisfies the homogeneous string equation $u_{tt}-u_{\tau \tau}=0$ on each edge $e$. Hence we see that the waves propagate from the boundary into $\Omega$ with unit velocity. It follows that if the control acts from a part $\Sigma\subseteq\Gamma$ of the boundary, i.e., $\operatorname{supp}f\subset\Sigma\times[0,T]$. then
$$
\begin{equation}
\operatorname{supp}u^f(\,{\cdot}\,,t) \subset \overline{\Omega^t_\Sigma}, \qquad t>0.
\end{equation}
\tag{2.6}
$$
In what follows we define (generalized) solutions of the problem (2.2)–(2.5) for controls of class $L_2(\Gamma \times [0,T])$. Their definition requires some preparation. We write $\delta_x$ for the Dirac measure, i.e., the functional on $C(\Omega)$ acting by the rule $\langle\delta_{x},y\rangle=y(x)$. Let $\delta=\delta(t)$ be the Dirac delta function. Our immediate goal is to define and describe the fundamental solution of the problem (2.2)–(2.5) with $T=\infty$, corresponding to the control $f=\delta_\gamma\delta(t)$, which acts instantaneously from a boundary vertex $\gamma$. To describe the solution $u^{\delta_\gamma \delta}$, it is convenient to use the following formalism of “impulse dynamics”. 0. By an impulse we mean the measure $a\delta_x$; the constant $a\ne 0$ is called its amplitude. 1. Each impulse $a\delta_{x(t)}$ moves along an edge with velocity $1$ in one of the two possible directions, so that $|\dot x(t)|=1$ for $x(t)\in e$. 2 (superposition principle). Impulses move independently of each other. If at the moment $t$ there are several impulses $a_1\delta_{x(t)},\dots,a_p\delta_{x(t)}$ located at the point $x(t)\in\Omega\setminus\Gamma$, they are combined to form the impulse $[\,a_1+\dots + a_p\,]\delta_{x(t)}$. 3 (passing through an interior vertex). Moving along an edge $e$ and passing through an interior vertex $v$, the impulse $a\delta_{x(t)}$ splits into $\mu(v)$ impulses: one reflected and $\mu(v)-1$ transmitted. The reflected impulse moves along $e$ in the opposite direction and has amplitude $(2-\mu(v)/\mu(v))a$. Each transmitted impulse moves along its own edge (incident to $v$) away from $v$ and has amplitude $(2/\mu(v))a$. Therefore, the total amplitude is
$$
\begin{equation*}
\frac{2-\mu(v)}{\mu(v)}\,a+[\mu(v)-1]\frac{2}{\mu(v)}\,a=a
\end{equation*}
\notag
$$
in accordance with Kirchhoff’s flow conservation law $F_v[y]=0$. 4 (reflection from the boundary). As soon as an impulse $a\delta_{x(t)}$ reaches a vertex $\gamma\,{\in}\,\Gamma$, it instantly inverts its direction and changes its amplitude from $a$ to $-a$. Adopting these rules, we can describe the solution $u^{\delta_\gamma\delta}$ as follows (recall that $\tau$ is the distance in $\Omega$). $(*)$ When $0\leqslant t\leqslant\tau(\gamma,V)$, we have $u^{\delta_\gamma\delta}=\delta_{x(t)}$, where $x(t)$ is the point of an edge $e\succ\gamma$ such that $\tau(x(t),\gamma)=t$. Thus, at small times, $u^{\delta_\gamma\delta}$ is a single impulse with unit amplitude entering the graph from the vertex $\gamma$ and moving along $e$ with unit velocity. $(**)$ Further evolution at times $t>\tau(\gamma,V)$ is determined by the rules 1–4. It is easy to see that this description is quite deterministic. At each moment of time $t\geqslant 0$, the solution $u^{\delta_\gamma\delta}$ represents a finite set of impulses moving in $\Omega$. From physical point of view, this picture describes, for example, the propagation of sharp signals (voltage spikes) in an electrical net (a graph made up of wires). Thus, the fundamental solution is a space-time distribution in $\Omega\times\{t\geqslant 0\}$. Its structure is such that the time convolution
$$
\begin{equation}
u^f(x,t):=\bigl[u^{\delta_\gamma\delta}(x,\,{\cdot}\,)\ast \varphi\bigr](t), \qquad x\in\Omega,\quad 0\leqslant t\leqslant T,
\end{equation}
\tag{2.7}
$$
is well defined for any control of the form $f=\delta_\gamma\varphi$ with $\varphi\in L_2[0,T]$. Moreover, it can be shown that $u^f\in C([0,T];\mathscr{H})$ and if $\varphi$ is smooth and vanishes around $t=0$, then $u^f$ coincides with the classical solution of the problem (2.2)–(2.5). From now on, we regard the function $u^f$ defined by the relation (2.7) as a (generalized) solution for controls of the form described above. In the more general case when $f\in L_2(\Gamma\times[0,T])$ is of the form $f=\sum_{\gamma\in\Sigma}f_\gamma$ with $f_\gamma=\delta_\gamma\varphi_\gamma$, we put
$$
\begin{equation}
u^f(x,t):=\sum_{\gamma\in\Sigma}u^{f_\gamma}(x,t), \qquad x\in\Omega,\quad 0\leqslant t\leqslant T.
\end{equation}
\tag{2.8}
$$
It is not difficult to show that the relation (2.6) remains valid for generalized solutions. Therefore the metric neighborhood $\Omega^T_\Sigma$ is the part of the graph captured at $t=T$ by waves coming from $\Sigma$. 2.3. HydrasHere we introduce a space-time graph which is used to efficiently describe waves. Fix a boundary vertex $\gamma$. Regarding the fundamental solution as a space-time distribution, we define the set
$$
\begin{equation*}
H_\gamma:=\operatorname{supp}u^{\delta_\gamma\delta}\subset \Omega\times{\overline{\mathbb R}}_+.
\end{equation*}
\notag
$$
We call it a hydra [6]. It is essentially the space-time graph formed by the impulse trajectories in the course of evolution described by the rules 1–4 and $(*)$, $(**)$ (Fig. 1).3 Its edges are characteristics of the wave equation (2.2). We define the projections
$$
\begin{equation*}
\begin{alignedat}{2} &\pi\colon H_\gamma \ni h=(x,t)\mapsto x\in \Omega, &\qquad \pi^{-1}(x) &:=\{h\in H_\gamma\mid \pi(h)=x\}; \\ &\rho\colon H_\gamma \ni h=(x,t)\mapsto t\in \overline{\mathbb R}_+, &\qquad \rho^{-1}(t) &:=\{h\in H_\gamma\mid \rho(h)=t\}. \end{alignedat}
\end{equation*}
\notag
$$
On the hydra, we define a function (amplitude) $a(\,{\cdot}\,)$ by the following rules. – If $h\in H_\gamma$ is such that $\pi(h)=x\in\Omega\setminus\Gamma$ and $\rho(h)=t>0$, then we have $u^{\delta_\gamma \delta}(\,{\cdot}\,,t)=a\delta_x(\,{\cdot}\,)$ and define $a(h):=a$. – If $h\in H_\gamma$ is such that $\pi(h)\in \Gamma$ and $\rho(h)>0$, we put $a(h):=0$. – If $h\in H_\gamma$ is such that $\pi(h)=\gamma$ and $\rho(h)=0$, we put $a(h):=1$. We see that the amplitude is a piecewise constant function defined on the whole hydra $H_\gamma$ (Fig. 2). Note that at self-intersection points $p$ we have $a(p)=-4/9+1/3=-1/9$ by rule 2 of impulse evolution. The self-intersection points like $p$ are the vertices of $H_\gamma$ projecting to $\Omega\setminus[V\cup\Gamma]$. Here is the representation for which the hydra was introduced (see [7]). Writing $h=(x,t)\in H_\gamma$ and $a(h)=a(x,t)$, we see from (2.7) that
$$
\begin{equation}
u^f(x,T)=\sum_{t\in \rho(\pi^{-1}(x))}a(x,t)\,\varphi(T-t),\qquad x\in\Omega,
\end{equation}
\tag{2.9}
$$
for controls of the form $f=\delta_\gamma\varphi(t)$ with $\varphi\in L_2[0,T]$. In the general case when the control $f=\sum_{\gamma\in\Sigma}\delta_\gamma\varphi_\gamma(t)$ acts from several vertices, we see from (2.8) that
$$
\begin{equation}
u^f(x,T)=\sum_{\gamma\in\Sigma}\,\sum_{t\in \rho(\pi^{-1}(x))}a_\gamma(x,t)\varphi_\gamma(T-t), \qquad x\in\Omega,
\end{equation}
\tag{2.10}
$$
where $a_\gamma$ is the amplitude on the hydra $H^T_\gamma$. These representations are quite efficient: they can be used to calculate the values of waves. However, the forthcoming analysis of the eikonal algebra requires their modification, which will be described now. It uses a partition of the graph $\Omega$ into parts (families) consistent with the structure of the hydra. This partition was described in [7] in full detail with graphic illustrations. 2.4. The partition $\Pi$In what follows we deal with truncated hydras We introduce the following general notion. Let $X$ be a set and let $\sim_0$ be a reflexive symmetric (but, generally speaking, not transitive!) binary relation on $X$. Elements connected by $\sim_0$ will be referred to as neighbors. This relation determines an equivalence relation by the following rule. We say that $x$ and $y$ are equivalent (and write $x\sim y$) if $X$ contains a finite set of elements $x_1,\dots,x_n$ such that $x\sim_0 x_1\sim_0 \dots\sim_0 x_n\sim_0 y$. The equivalence class $[x]$ of an element $x\in X$ with respect to $\sim$ can be described constructively. We define an operation $\operatorname{ext}$ extending any subset $B\subset X$ by the rule
$$
\begin{equation}
B\mapsto \operatorname{ext}B:=\bigcup_{b\in B}\{x\in X\mid x\sim_0 b\}.
\end{equation}
\tag{2.11}
$$
Put $\operatorname{ext}^1B:=\operatorname{ext}B$ and $\operatorname{ext}^jB:=\operatorname{ext}\operatorname{ext}^{j-1}B$, $j\geqslant 2$. Then it is easy to see that the following representation holds:
$$
\begin{equation}
[x]=\bigcup_{j\geqslant 1}\operatorname{ext}^j\{x\}.
\end{equation}
\tag{2.12}
$$
If $X$ is finite, the sequence $\operatorname{ext}^j$ stabilizes: $\operatorname{ext}^1\{x\}\subset\dots\subset \operatorname{ext}^N\{x\}=\operatorname{ext}^{N+1}\{x\}=\dots=[x]$. This is the case in the present paper. Consider an equivalence of this kind on the hydra. We say that points $h,h'\in H^T_\gamma$ are neighbors ($h\overset{\gamma}\sim_0 h' $) if at least one of the following conditions holds: $\pi(h)=\pi(h')$ or $\rho(h)=\rho(h')$. We write $\stackrel{\gamma}{\sim}$ for the equivalence relation generated by this notion of neighboring. Every equivalence class
$$
\begin{equation*}
\mathscr{L}[h]:=\{h'\in H^T_\gamma\mid h'\stackrel{\gamma}{\sim} h\}
\end{equation*}
\notag
$$
is called a lattice. One can show that it consists of a finite number of points. For any subset $B\subset H^T_\gamma$ we define the lattice Note that the operation $B\mapsto \mathscr{L}[B]$ has the following properties:
$$
\begin{equation*}
\begin{gathered} \, B\subset \mathscr{L}[B],\qquad \mathscr{L}[\mathscr{L}[B]]=\mathscr{L}[B],\qquad \mathscr{L}[B_1\cup B_2]=\mathscr{L}[B_1]\cup\mathscr{L}[B_2], \\ \pi^{-1}(\pi(\mathscr{L}[B]))=\rho^{-1}(\rho(\mathscr{L}[B]))=\mathscr{L}[B]. \end{gathered}
\end{equation*}
\notag
$$
The first three properties show that it is a topological closure (in the sense of Kuratowski). With every point $x\in\overline{\Omega^T_\gamma}$ we associate the set
$$
\begin{equation}
\Lambda[x]:=\pi(\mathscr{L}[\pi^{-1}(x)])\subset \overline{\Omega^T_\gamma}
\end{equation}
\tag{2.13}
$$
(the closure in the metric of $\Omega$). We call it the determination set of $x$. This set is finite. It is easy to check that the relation
$$
\begin{equation*}
x\sim x'\quad \Longleftrightarrow\quad \Lambda[x]=\Lambda[x']
\end{equation*}
\notag
$$
is an equivalence, and the operation $A\mapsto\Lambda[A]:=\bigcup_{x\in A}\Lambda[x]=\pi(\mathscr L[\pi^{-1}(A)])$ is a topological closure. The sets with $A=\Lambda[A]$ are said to be $\Lambda$-closed. A point $h\in H_\gamma$ on a complete hydra is called a corner point if $\pi(h)\in V\cup\Gamma$ or $h$ is a self-intersection point (like $p$ on Fig. 2). The latter points are the vertices of valence 4 projecting to $\Omega\setminus[V\cup\Gamma]$. On the truncated hydra $H^T_\gamma$, besides the corner points of the complete hydra, we declare the points in $\rho^{-1}(T)$ to be corner points. The set of all corner points of the truncated hydra is denoted by $\operatorname{Corn}H^T_\gamma$. The lattice $\mathscr{L}[\operatorname{Corn} H^T_\gamma]$ divides the hydra into a finite number of open space-time intervals. The amplitude $a$ is constant on each interval. The points in the finite set
$$
\begin{equation*}
\Theta:=\pi\bigl(\mathscr{L}[\operatorname{Corn}H^T_\gamma]\bigr)\subset \overline{\Omega^T_\gamma}
\end{equation*}
\notag
$$
are said to be critical. All other points $x\in \overline{\Omega^T_\gamma}\setminus\Theta$ are said to be regular. Critical points divide $\overline{\Omega^T_\gamma}$ into parts. The set of regular points is the union of finitely many open intervals, each of which lies on some edge $e$. Thus, $\overline{\Omega^T_\gamma}=\Pi\cup\Theta$ is a partition (determined by the hydra structure $H^T_\gamma$) of the part of $\Omega$ captured by waves. Let $\omega=(c,c')\subset \Pi$ be a maximal interval consisting of regular points. Maximality of $\omega$ means that its ends $c$ and $c'$ are critical points, so that it is impossible to extend $\omega$ keeping its interior points regular. It is easy to see that the set
$$
\begin{equation}
\Phi:=\Lambda(\omega)=\pi(\mathscr{L}[\pi^{-1}(\omega)])
\end{equation}
\tag{2.14}
$$
consists of maximal intervals $\omega_1,\dots,\omega_m$ of equal length:
$$
\begin{equation*}
\Phi=\bigcup_{k=1}^m \omega_k, \qquad \operatorname{diam}\omega_1=\dots=\operatorname{diam}\omega_m=\tau(c,c')=:\epsilon_\Phi,
\end{equation*}
\notag
$$
where $\tau$ is the distance in the graph. We say that the intervals $\omega_k$ are cells of the family $\Phi$. Comparing the definitions (2.13) and (2.14), we arrive at the representation where $\omega$ is any of the cells of $\Phi$.Let $\omega'\subset\Pi$ be a maximal interval not lying in the family $\Phi$. It determines another family $\Phi'=\Lambda[\omega']$ consisting of cells, and so on. As a result, $\Pi$ is the finite union of disjoint families $\Phi^1,\dots,\Phi^J$, each of which consists of disjoint cells:
$$
\begin{equation}
\Pi=\bigcup_{j=1}^J\Phi^j=\bigcup_{j=1}^J\bigcup_{k=1}^{m_j}\omega^j_k,
\end{equation}
\tag{2.16}
$$
where $m_j$ is the number of cells in $\Phi^j$. In parallel to the determination set (2.13), each $x\in\overline{\Omega^T_\gamma}\setminus\Gamma$ is associated with the set For $x\ne x'$ we have either $\Xi[x]=\Xi[x']$ or $\Xi[x]\cap\Xi[x']=\varnothing$. We also define $\Xi[B]:=\bigcup_{x\in B}\Xi[x]$.Consider a family $\Phi=\bigcup_{k=1}^{m_\Phi}\omega_k\subset\Pi$. It is easy to check that the set
$$
\begin{equation}
\Psi:=\Xi[\Phi]=\bigcup_{i=1}^{n_\Phi}\psi_i\subset [0,T]
\end{equation}
\tag{2.17}
$$
consists of time intervals $\psi_i: =(t_i,\widetilde{t}_i)$ satisfying $0\leqslant t_1<\widetilde{t}_1\leqslant t_2<\widetilde{t}_2\leqslant \dots\leqslant t_{n_\Phi }<\widetilde{t}_{n_\Phi}\leqslant T$ and having equal lengths $\widetilde{t}_i-t_{i}=\epsilon_\Phi$. The set $\Psi$ will also be called a family consisting of the time cells $\psi_i$. We shall use the functions $\tau^i\colon \Phi \to [0,T]$ associated with the partition of the graph into families. They are introduced as follows.4 For $x\in\Phi$, we set
$$
\begin{equation}
\tau^i(x):=\psi_i\cap \rho(\mathscr{L}[\pi^{-1}(x)]),\qquad i=1,\dots, n_\Phi.
\end{equation}
\tag{2.18}
$$
Since $\mathscr{L}[\pi^{-1}(x)]=\mathscr{L}[\pi^{-1}(x_k)]$ for any $x_k\in\Lambda[x]$, these functions are constant on the determination sets: $\tau^i(x)=\tau^i(x_k)$. It follows from the definition that
$$
\begin{equation}
\tau^i(x)\ne \tau^{i'}(x)\quad\text{for}\quad i\ne i',\ \ x\in\Phi.
\end{equation}
\tag{2.19}
$$
When $x$ varies inside the cell $\omega=(c,c')\subset\Phi$, the set $\Lambda[x]$ changes. The possible values of $\tau^i(x)$ sweep out the cell $\psi_i=(t_i,\widetilde t_i)\subset\Psi$ and we easily see from the definition (2.18) that one of the following two representations holds:
$$
\begin{equation}
\tau^i(x)=t_i+\tau(x,c)\quad \text{or}\quad \tau^i(x)=\widetilde t_i-\tau(x,c).
\end{equation}
\tag{2.20}
$$
With this representation in mind, we can say that the functions $\tau^i$ depend linearly on $x\in\omega$. The property (2.20) enables us to extend the functions $\tau^i$ to critical points. Namely, if $x\in\omega=(c,c')$, $x\to c$, then $\tau^i(c)=t_i$ or $\tau^i(c)=\widetilde t_i$ depending on which of the representations (2.20) holds. For each family $\Phi\subset\Pi$, we have its own functions $\tau^i$ and we denote them by $\tau^i_\Phi$ if necessary. Moreover, by (2.19) and (2.20), the equality $\tau^i_\Phi(x)=\tau^{i'}_{\Phi'}(x)$ for different $\Phi$ and $\Phi'$ is possible only at critical points $x$. Our partition of the graph into families is motivated, in particular, by the fact that the waves $u^f$ depend on the controls $f$ locally in the following sense. We see from (2.9) that the values of $u^f(\,{\cdot}\,,T)|_{\Phi}$ are determined by the values of $f|_{\Xi[\Phi]}$. Moreover, the conditions
$$
\begin{equation}
\operatorname{supp}f\subset \Xi[\Phi]\quad\text{and}\quad\operatorname{supp}u^f(\,{\cdot}\,,T)\subset \Phi
\end{equation}
\tag{2.21}
$$
are equivalent. The modification of (2.9) and (2.10), mentioned at the end of the previous subsection, consists in an efficient representation of waves on determination sets. It uses functions (vectors) on $\Lambda[x]$ which will be described now. 2.5. Amplitude vectorsWe still fix a boundary vertex $\gamma\in\Gamma$, a number $T>0$ and the corresponding partition $\Pi$. Suppose that $A=\{x_1,\dots,x_m\}\subset\overline{\Omega^T_\gamma}$ is a finite $\Lambda$-closed set and
$$
\begin{equation*}
\Xi[A]=\rho(\mathscr L[\pi^{-1}(A)])=\{t_1,\dots, t_n\},
\end{equation*}
\notag
$$
where $t_1<\dots<t_n$. We associate with $A$ a set of functions (amplitude vectors) $\alpha^i\colon A\to\mathbb{R}$, $i=1,\dots,n$, by the following rule:
$$
\begin{equation*}
\alpha^i(x_k):=\begin{cases} a(x_k,t_i) &\text{if }(x_k,t_i)\in H^T_\gamma, \\ 0 &\text{if }(x_k,t_i)\notin H^T_\gamma, \end{cases}\qquad k=1,\dots,m.
\end{equation*}
\notag
$$
The set $\{\alpha^1,\dots,\alpha^n\}$ will be called the $\alpha$-set over $A$. For each $x\in\in\overline{\Omega^T_\gamma}$, the determination set $\Lambda[x]$ is $\Lambda$-closed and there is an $\alpha$-set over it. It follows from (2.6) that the value is the time of filling the whole graph $\Omega$ with waves outgoing (with unit velocity) from the vertex $\gamma$.Lemma 1. Let $x$ be a point in $\overline{\Omega^T_\gamma}$, $\Lambda[x]$ its determination set, and $\alpha^1,\dots,\alpha^n$ the $\alpha$-set over $\Lambda[x]$. If $T<T_\gamma$, then the vectors $\alpha^i$ are linearly independent. Proof. Fix a number $t^*=t_i\in\Xi[x]$ with $i\neq 1$. The lattice $\mathscr{L}[\pi^{-1}(x)]$ necessarily contains a point $(x^*,t^*)$ such that $x^*$ belongs to the boundary $\overline{\Omega^{t^*}_\gamma}\setminus{\Omega^{t^*}_\gamma}$ of the domain ${\Omega^{t^*}[\gamma]}$ captured by waves at the moment $t=t^*$. This boundary is non-empty since $t^*\leqslant T<T_\gamma$. Moreover, $x^*\in\Lambda[x]$, $t^*=\tau(x^*,\gamma)$ and, clearly, we have $a(x^*,t^*)=\alpha^i(x^*)\ne 0$.
At the same time, $a(x^*,t)=0$ for all points $(x^*,t)$ with $t\in\Xi[x^*]=\Xi[x]$, $t<t^*$, because these points $(x^*,t)$ do not lie on the hydra $H^T_\gamma$. This corresponds to the simple fact that at the given times $t$ the waves from $\gamma$ do not have time to reach the point $x^*$ (see (2.6)). Thus, at points $(x^*,t)$ with $t=t_k\in\Xi[x]$, $t_i<t^*$ ($k=1,\dots,{i-1}$) we have $a(x^*,t_k)=\alpha^k(x^*)=0$ and, at the same time, $a(x^*,t^*)=\alpha^i(x^*)\ne 0$. Hence $\alpha^i$ cannot be a linear combination of $\alpha^1,\dots,\alpha^{i-1}$. Since $i$ is arbitrary, it follows that the whole set $\alpha^1,\dots,\alpha^n$ is linearly independent. $\Box$ Indeed, we see from the proof of Lemma 1 that if $T<T_\gamma$, then the set $\rho^{-1}(t)\subset H^T_\gamma$ is non-empty for $0\leqslant t\leqslant T$, which is equivalent to the equality $\Xi[\overline{\Omega^T_\gamma}]=[0,T]$. We now reconsider the representation (2.9). In terms of amplitude vectors, it can be written in the form
$$
\begin{equation}
u^f(x_k,T)|_{x_k\in\Lambda[x]}=\sum_{i=1}^{n[x]}\varphi(T-t_i)\alpha^i(x_k),\quad \text{for}\quad f=\delta_\gamma\varphi,\ \ n[x]:=\sharp\,\Xi[x],
\end{equation}
\tag{2.22}
$$
representing the wave not only at point $x$ but also on the whole determination set $\Lambda[x]$. Moreover, according to (2.15), by varying the point $x$ inside the cell $\omega\subset\Phi$, we represent the wave $u^f(\,{\cdot}\,,T)$ on the whole family $\Phi$. The final step of modification of the original representation (2.9) consists in passing to a more convenient system of amplitude vectors in (2.22). 2.6. The $\beta$-representation of wavesAgain, let $A\,{=}\,\Lambda[A]\,{=}\,\{x_1,\dots,x_m\}\,{\subset}\, \overline{\Omega^T_\gamma}$. We introduce the space $\mathbf{l}_2(A)$ of functions (vectors) on $A$ with scalar product
$$
\begin{equation*}
\langle f,g\rangle=\sum_{x\in A}f(x)g(x)=\sum_{k=1}^{m} f(x_k)g(x_k).
\end{equation*}
\notag
$$
It contains a subspace
$$
\begin{equation*}
\mathbb{A}[A]:=\operatorname{span}\{\alpha^1,\dots,\alpha^{n}\},\qquad \operatorname{dim}\mathbb{A}[A]\leqslant n,
\end{equation*}
\notag
$$
which is spanned by the $\alpha$-set over $A$. Using the Gram–Schmidt procedure, we pass in $\mathbb{A}[A]$ to the set
$$
\begin{equation*}
\beta^i:= \begin{cases} \dfrac{\alpha^1}{\|\alpha^1\|} &\text{if }i=1, \\ \dfrac{\alpha^i-\sum _{j=1}^{i-1}\langle \alpha^i,\beta^j\rangle \beta^j}{\|\alpha^i-\sum_{j=1}^{i-1}\langle \alpha^i,\beta^j\rangle \beta^j\|} &\text{if }i\geqslant 2\text{ and }\alpha^i \notin \operatorname{span}\{\alpha^1, \dots, \alpha^{i-1}\}, \\ 0 &\text{if }\alpha^i \in \operatorname{span}\{\alpha^1, \dots, \alpha^{i-1}\}, \end{cases}
\end{equation*}
\notag
$$
where $\beta^i$ is the vector with components $\beta^i(x_1),\dots,\beta^i(x_m)$. We have $\langle \beta^i,\beta^j \rangle=\delta_{ij}$ for non-zero elements of this set, and their linear span obviously coincides with $\mathbb{A}[A]$. The set of vectors $\{\beta^1,\dots,\beta^n\}$ will be called the $\beta$-set over $A$. For every $x\in\overline{\Omega^T_\gamma}$ we have its own $\beta$-set over the determination set $\Lambda[x]$. By Lemma 1, the vectors $\alpha^i$ are linearly independent if $T<T_\gamma$. Hence, for these $T$, the vectors $\beta^i$ are all non-zero and $\operatorname{dim}\mathbb{A}[\Lambda[x]]=n\leqslant m$. In terms of the $\beta$-set over the set $A=\Lambda[x]$, the representation (2.22) takes its final form
$$
\begin{equation}
u^f(x_k,T)|_{x_k\in\Lambda[x]}=\sum_{j=1}^{n[x]}c^\varphi_j\beta^j(x_k),\qquad c^\varphi_j=\sum_{i=1}^{n[x]}\rho_{ji}\,\varphi(T-t_i),
\end{equation}
\tag{2.23}
$$
where $\rho$ is the transition matrix connecting the sets $\alpha$ and $\beta$. 2.7. The hydra $H^T_\Sigma$The concepts and objects introduced above correspond to a single boundary vertex $\gamma$. In what follows, we use the notation $\overset{\gamma}\sim$, $\mathscr L_\gamma$, $\Lambda_\gamma$, etc., to indicate this if necessary. Given any set $\Sigma\subseteq\Gamma$ of boundary vertices, we define a space-time graph
$$
\begin{equation*}
H^T_\Sigma:=\bigcup_{\gamma\in\Sigma}H^T_\gamma\subset\Omega\times[0,T].
\end{equation*}
\notag
$$
It carries analogs of the objects introduced earlier for the single hydras $H^T_\gamma$. We proceed to describe them. The projections from $H^T_\Sigma$ to $\overline{\Omega^T_\Sigma}$ and $[0,T]$ are $\pi((x,t)):=x$ and $\rho((x,t)):=t$. By $\pi^{-1}$ and $\rho^{-1}$ we mean the full preimages in $H^T_\Sigma$. By definition, we write $h\sim_0 h'$ on $H^T_\Sigma$ if $\pi(x)=\pi(x')$ and/or $\rho(x)=\rho(x')$. This relation of neighboring determines an equivalence relation $h\overset{\Sigma}\sim h'$. We denote the equivalence class (lattice) of a point $h\in H^T_\Sigma$ by $\mathscr L_\Sigma[h]$. The operation $H^T_\Sigma\supset B\mapsto \mathscr L_\Sigma[B]$ is a (topological) closure. The set $\Lambda_\Sigma[x]:=\pi(\mathscr{L}_\Sigma[\pi^{-1}(x)])$ will be called the determination set of $x\in\overline{\Omega^T_\Sigma}$. Note the obvious embedding $\Lambda_\gamma[x]\subset\Lambda_\Sigma[x]$ for $\gamma\in\Sigma$. The operation $\overline{\Omega^T_\Sigma}\supset A\mapsto \Lambda_\Sigma[A]$ is a (topological) closure. The set of corner points $\operatorname{Corn}H^T_\Sigma$ consists of all corner points of the hydras $H^T_\gamma\subset H^T_\Sigma$ plus the (transversal) intersection points of the edges of different hydras $H^T_\gamma$. The critical points in $\overline{\Omega^T_\Sigma}$ are $\Theta_\Sigma:=\pi(\mathscr L_\Sigma[\operatorname{Corn}H^T_\Sigma])$, the regular points are $\Pi_\Sigma:=\overline{\Omega^T_\Sigma}\setminus\Theta_\Sigma$. There is a partition
$$
\begin{equation*}
\Pi_\Sigma=\bigcup_{j=1}^J\Phi^j=\bigcup_{j=1}^J\bigcup_{k=1}^{m_j}\omega^j_k,\qquad \operatorname{diam}\omega^j_k=\epsilon_j:=\epsilon_\Phi,
\end{equation*}
\notag
$$
into families and cells, which is quite similar to the partition (2.16). For each family $\Phi\subset\Pi_\Sigma$, the set
$$
\begin{equation*}
\Xi_\Sigma[\Phi]:=\rho(\mathscr L_\Sigma[\pi^{-1}(\Phi)]) =\bigcup_{i=1}^{n_\Phi}\psi_i\subset [0,T]
\end{equation*}
\notag
$$
consists of time intervals $\psi_i=(t_i,\widetilde{t}_i)$ such that $0\leqslant t_1<\widetilde{t}_1\leqslant t_2<\widetilde{t}_2\leqslant \dots \leqslant t_{n_\Phi }<\widetilde{t}_{n_\Phi}\leqslant T$; all intervals have the same length $\widetilde{t}_i-t_{i}=\epsilon_\Phi$. On each family, we define a set of functions $\tau^i$ by putting
$$
\begin{equation}
\tau^i_\Phi(x):=\psi_i\cap \rho(\mathscr{L}_\Sigma[\pi^{-1}(x)]),\qquad x\in\Phi,\quad i=1,\dots, n_\Phi.
\end{equation}
\tag{2.24}
$$
These functions are constant on determination sets: $\tau^i_\Phi(x)=\tau^i_\Phi(x')$ for regular $x,x'\in \Lambda_\Sigma[x]$. They have the property (2.19) and satisfy the representation (2.20). This enables us to extend the functions $\tau^i_\Phi$ to the critical points (the ends of the cells $\omega$) by continuity. Suppose that $\gamma\in\Sigma$, $\Phi\subset\Pi_\Sigma$ and $x\in \Phi$. We easily see that the set $\Lambda_\Sigma[x]\cap {\Omega^T_\gamma}$ is $\Lambda_\gamma$-closed (in $\Omega^T_\gamma$). Hence it has a $\beta$-set of vectors $\beta^1_{\gamma\Phi},\dots, \beta^{n_\Phi}_{\gamma\Phi}$. We define them on the whole $\Lambda_\Sigma[x]$, extending them from $\Lambda_\Sigma[x]\cap \Omega^T_\gamma$ to $\Lambda_\Sigma[x]$ by zero. Repeating the construction for all $\gamma\in\Sigma$, we obtain the $\beta$-sets
$$
\begin{equation}
\{\beta^1_{\gamma\Phi},\dots,\beta^{n_{\Phi}}_{\gamma\Phi}\mid\gamma\in\Sigma\},
\end{equation}
\tag{2.25}
$$
where $\beta^i_{\gamma\Phi}$ is the vector with components $(\beta^i_{\gamma\Phi})_1,\dots,(\beta^i_{\gamma\Phi})_{m_{\Phi}}$ and $m_{\Phi}=\sharp\Lambda_\Sigma[x]$. Each of the sets is orthonormal in $\mathbf{l}_2(\Lambda_\Sigma[x])$. Accordingly, on each family $\Phi\subset\Pi_\Sigma$ there are functions $\beta^i_{\gamma\Phi}(\,{\cdot}\,)$, which take constant values $(\beta^i_{\gamma\Phi})_k$ on the cells $\omega_k\subset\Phi$. For a given family $\Phi$ and different vertices $\gamma\in\Sigma$, the set $\{\beta^1_{\gamma\Phi},\dots,\beta^{n_{\Phi}}_{\gamma\Phi}\}$ contains the same number of vectors (namely, $n_{\Phi}$) and the functions $\tau^i_\Phi$ are the same. Nevertheless, we assume by definition that
$$
\begin{equation}
n_{\gamma\Phi}:=n_{\Phi},\qquad \tau_{\gamma\Phi}^i(x):=\tau_{\\Phi}^i(x),\qquad x\in\Phi, \quad \gamma\in\Sigma.
\end{equation}
\tag{2.26}
$$
This seemingly redundant notation (the subscript $\gamma$) will prove convenient in what follows.
§ 3. Eikonals3.1. Reachable sets and projectorsHere we consider the problem (2.2)–(2.5) as a dynamical system and equip it with control theory attributes (spaces and operators). The space of controls $\mathscr{F}^T :=L_2(\Gamma \times [0,T])$ with the inner product
$$
\begin{equation*}
(f,g)_{\mathscr{F}^T}=\sum_{\gamma \in \Gamma} \int_{0}^T f(\gamma,t)g(\gamma,t)\,dt
\end{equation*}
\notag
$$
is called the external space of the system (2.2)–(2.5). It contains subspaces of the controls acting from separate boundary vertices $\gamma \in \Gamma$:
$$
\begin{equation*}
\mathscr{F}^T_\gamma:=\bigl\{f \in \mathscr{F}^T\bigm| \operatorname{supp}f \subset \{\gamma\} \times [0,T]\bigr\}
\end{equation*}
\notag
$$
Each control $f \in \mathscr{F}^T_\gamma$ is of the form $f=\delta_{\gamma}\varphi$ for some $\varphi\in L_2[0,T]$. For every set $\Sigma\subseteq\Gamma$ of boundary vertices we have the subspace
$$
\begin{equation*}
\mathscr{F}^T_\Sigma:={\sum_{\gamma \in \Sigma}}^\oplus \mathscr{F}^T_\gamma
\end{equation*}
\notag
$$
(the summands are orthogonal in $\mathscr{F}^T$). The space $\mathscr{H}=L_2(\Omega)$ is said to be internal; the waves $u^f(\,\cdot\,,t)$ can be regarded as evolutions of its elements in time. For every set $B\subset\Omega$ we define the subspace $\mathscr H\langle B\rangle:=\{y\in\mathscr H\mid \operatorname{supp}y\subset\overline B\}$ of functions localized in $B$. The set of waves
$$
\begin{equation*}
\mathscr{U}^s_\gamma:=\{u^f(\,{\cdot}\,,s)\mid f \in \mathscr{ F}^T_\gamma\}\subset \mathscr{H}, \qquad 0\leqslant s\leqslant T,
\end{equation*}
\notag
$$
is said to be reachable (from the vertex $\gamma$ at the moment $t=s$). It follows from the representations (2.9) and (2.10) that the sets $\mathscr{U}^s_\gamma$ are (closed) subspaces in $\mathscr H$. They expand as $s$ grows: $\mathscr{U}^s_\gamma\subset\mathscr{U}^{s'}_\gamma$ if $s<s'$. The locality of the “control-wave” correspondence noted in (2.21) leads to a decomposition into families:
$$
\begin{equation}
\mathscr{U}^T_\gamma={\sum_{\Phi\subset\Pi_\Sigma}}^\oplus \mathscr{U}^T_\gamma\langle\Phi\rangle,
\end{equation}
\tag{3.1}
$$
where the subspace $\mathscr{U}^T_\gamma\langle\Phi\rangle\subset\mathscr H\langle\Phi\rangle$ consists of the waves $u^f(\,{\cdot}\,,T)\in\mathscr U^T_\gamma$ localized in $\Phi\,\cap\,\Omega^T_\gamma$. The summands are orthogonal since the families are disjoint: $\Phi^j\,\cap \,\Phi^k= \varnothing$ for $j\ne k$. We fix a boundary vertex $\gamma\in\Sigma$. Let $P^T_{\gamma}$ be the projector in $\mathscr{H}$ to the subspace $\mathscr{U}^T_\gamma$. We shall discuss its properties and describe its action. By (3.1), we have a representation
$$
\begin{equation}
P^T_\gamma=\sum _{\Phi \subset \Pi_\Sigma}P^T_{\gamma}\langle\Phi\rangle,
\end{equation}
\tag{3.2}
$$
where $P^T_{\gamma}\langle\Phi\rangle$ is the projector in $\mathscr{H}$ to $\mathscr{U}^T_\gamma\langle\Phi\rangle$. Thus, the projector $P^T_\gamma$ is reduced by the subspaces $\mathscr{U}^T_\gamma\langle\Phi\rangle$ and to characterize it, one needs to describe the action of $P^T_{\gamma}\langle\Phi\rangle$. It was shown in [7] that the projectors $P^T_{\gamma}\langle\Phi\rangle$ can be expressed in terms of the vectors (2.25) and the corresponding functions $\beta^i_{\gamma\Phi}(\,{\cdot}\,)$ as follows:
$$
\begin{equation}
(P^T_{\gamma}\langle\Phi\rangle y)(x)= \begin{cases} {\displaystyle\sum _{i=1}^{n_{\gamma\Phi}} \langle y|_{\Lambda_\gamma[x]}, \beta^i_{\gamma\Phi}\rangle \beta^i_{\gamma\Phi}(x)}, &x \in \Phi, \\ 0, &x \in \Omega \setminus\Phi, \end{cases}
\end{equation}
\tag{3.3}
$$
where $y \in \mathscr H$ is an arbitrary function on the graph.5 This representation is derived from the expression (2.23), which was the reason for introducing the vectors $\beta^i$. We see from (3.2) and (3.3) that if the function $y\in\mathscr{H}$ to be projected to $\mathscr{U}^T_\gamma$ is continuous and $y|_{\Lambda_\gamma[x]}\equiv 0$, then $(P^T_\gamma y)|_{\Lambda_\gamma[x]}\equiv 0$. In other words, the values of $P^T_\gamma y$ on $\Lambda_\gamma[x]$ are completely determined by the values of $y$ on $\Lambda_\gamma[x]$. This is the reason why the sets $\Lambda_\gamma[x]$ as well as $\Lambda_\Sigma[x]$ are called determination sets. Recall that the vectors $\beta^i_{\gamma\Phi}$ are elements of the subspace $\mathbb A[\Lambda_\Sigma[x]]\subset\mathbf{l}_2(\Lambda_\Sigma[x])$. It follows from the above that the projector $P^T_\gamma$ determines in $\mathbf{l}_2(\Lambda_\Sigma[x])$ the operator $p_{\gamma\Phi}[x]$ which is the projector onto $\mathbb A[\Lambda_\gamma[x]]\subset\mathbb A[\Lambda_\Sigma[x]]$. Let $\chi_1,\dots, \chi_{m_\Phi}$ be the (standard) basis in $\mathbf{l}_2(\Lambda_\Sigma[x])$ consisting of the indicators of points of the set $\Lambda_\Sigma[x]$. In this basis, we have $(\beta^i_{\gamma\Phi})_k=\langle\beta^i_{\gamma\Phi},\chi_k\rangle$. By (3.3), the matrix of the projector $p_{\gamma\Phi}[x]$ takes the following form in this basis:
$$
\begin{equation}
{\check p}_{\gamma\Phi}[x]=B^*_{\gamma\Phi}[x]B_{\gamma\Phi}[x],\qquad B_{\gamma\Phi}[x]:= \begin{pmatrix} (\beta^1_{\gamma\Phi})_1(x) & \dots &(\beta^1_{\gamma\Phi})_{m_{\Phi}}(x) \\ (\beta^{2}_{\gamma\Phi})_1(x) & \dots & (\beta^{2}_{\gamma\Phi})_{m_{\Phi}}(x) \\ \dots & \dots & \dots \\ (\beta^{n_{\gamma\Phi}}_{\gamma\Phi})_1(x) & \dots & (\beta^{n_{\gamma\Phi}}_{\gamma\Phi}(x)) \end{pmatrix} .
\end{equation}
\tag{3.4}
$$
3.2. EikonalsThe family $\{P^s_\gamma\mid 0\leqslant s\leqslant T\}$ of the projectors in $\mathscr{H}$ to the reachable sets $\mathscr{U}^s_\gamma$ determines the eikonal operator (briefly, the eikonal)
$$
\begin{equation*}
E^T_\gamma\colon\mathscr{H}\to \mathscr{H},\qquad E^T_\gamma:=\int^T_0 s\,dP^s_\gamma.
\end{equation*}
\notag
$$
It follows from the definition that $E^T_\gamma$ is a bounded self-adjoint positive operator. Like the projector $P^T_\gamma$, the eikonal $E^T_\gamma$ is reduced by the subspaces $\mathscr{H}\langle\Phi\rangle$. For this operator, we have an inclusion $E^T_\gamma\mathscr{H}\langle\Phi\rangle\subset\mathscr{H}\langle\Phi\rangle$ and an expansion
$$
\begin{equation*}
E^T_\gamma=\sum_{\Phi\subset\Pi}E^T_\gamma\langle\Phi\rangle,
\end{equation*}
\notag
$$
where $E^T_\gamma\langle\Phi\rangle:=E^T_\gamma|_{\Phi}$ is the part of $E^T_\gamma$ acting in $\mathscr{H}\langle\Phi\rangle$. It was shown in [7] that we have a representation
$$
\begin{equation}
(E^T_\gamma\langle\Phi\rangle y)(x)= \begin{cases} {\displaystyle\sum _{i=1}^{n_{\gamma\Phi}} \tau^i_{\gamma\Phi}(x)\langle y|_{\Lambda_\gamma[x]},\beta^i_{\gamma\Phi}\rangle \beta^i_{\gamma\Phi}(x)}, &x \in \Phi, \\ 0, &x \in \Omega \setminus\Phi, \end{cases}
\end{equation}
\tag{3.5}
$$
which is consistent with (3.3). Here $y \in \mathscr H$ is arbitrary and the functions $\tau^i_{\gamma\Phi}$ are given by the definitions6 (2.24) and (2.26). The operator $E^T_\gamma$ induces the operator $e_{\gamma\Phi}[x]$ in $\mathbf{l}_2(\Lambda_\Sigma[x])$. By (3.4) and (3.5), its matrix in the basis $\chi_1,\dots, \chi_{m_\Phi}$ is of the form
$$
\begin{equation*}
\begin{gathered} \, {\check e}_{\gamma\Phi}[x]=B^*_{\gamma\Phi}[x]D_{\gamma\Phi}[x] B_{\gamma\Phi}[x], \\ B_{\gamma\Phi}[x]:= \begin{pmatrix} (\beta^1_{\gamma\Phi})_1(x) & \dots &(\beta^1_{\gamma\Phi})_{m_{\Phi}}(x) \\ (\beta^{2}_{\gamma\Phi})_1(x) & \dots & (\beta^{2}_{\gamma\Phi})_{m_{\Phi}}(x) \\ \dots & \dots & \dots \\ (\beta^{n_{\gamma\Phi}}_{\gamma\Phi})_1(x) & \dots & (\beta^{n_{\gamma\Phi}}_{\gamma\Phi})_{m_{\Phi}}(x) \end{pmatrix},\qquad D_{\gamma\Phi}[x]=\operatorname{diag}\{\tau^i_{\gamma\Phi}(x)\}_{i=1}^{n_{\gamma\Phi}}. \end{gathered}
\end{equation*}
\notag
$$
This matrix is reduced by the subspace $\mathbb A_{\gamma\Phi}[x]=\operatorname{span}\{\beta^1_{\gamma\Phi}. \dots, \beta^{n_{\gamma\Phi}}_{\gamma\Phi}\}\subset \mathbf{l}_2(\Lambda_\Sigma[x])$. Its non-zero block in the basis $\beta^1_{\gamma\Phi}, \dots, \beta^{n_{\gamma\Phi}}_{\gamma\Phi}$ is $\operatorname{diag}\{\tau^i_{\gamma\Phi}(x)\}_{i=1}^{n_{\gamma\Phi}}$. We see from (2.19) that all its eigenvalues $\tau^i_{\gamma\Phi}(x)$ on the cells of the family are distinct. When $x$ varies in $\omega\subset\Phi$, they fill the intervals $(t_i,\widetilde t_i)\subset\Xi[\Phi]$. Proposition 1 below shows that the union (over all families $\Phi\subset\Pi$) of the intervals $[t_i,\widetilde t_i]$ is equal to $[0,T]$ if $T<T_\gamma$. By what was said above, we easily obtain some general properties of eikonals as operators in $\mathscr{H}$ (see, e.g., [23]). Proposition 1. For the eikonal $E^T_\gamma$ we have $\overline{\operatorname{Ran}E^T_\gamma}\,{=}\, \mathscr{U}^T_\gamma$ and $\operatorname{Ker}E^T_\gamma\,{=}\,\mathscr{H}\,{\ominus}\, \mathscr{U}^T_\gamma$. The eikonal is reduced by the parts of the reachable set: $E^T_\gamma\mathscr{U}^T_\gamma\langle\Phi\rangle \subset\mathscr{U}^T_\gamma \langle\Phi\rangle$, $\Phi\subset\Pi_\Sigma$. If $T<T_\gamma$, then the operator $E^T_\gamma|_{\mathscr{U}^T_\gamma}$ has a simple absolutely continuous spectrum filling the interval $[0,T]$. Remark 1. It follows from the representation (3.5) that if $T>T_\gamma$, then the spectrum of $E^T_\gamma|_{\mathscr{U}^T_\gamma}$ is the union $[0,T_0]\cup[T_1,T_2]\cup\dots \cup[T_{N-1},T_N]$ of closed intervals, where $T_\gamma\leqslant T_0<T_1<\dots <T_N\leqslant T$ and each interval consists of the ranges of the functions7 $\tau^i_{\gamma\Phi}$ (the closures of the time cells ${\psi^i_{\gamma\Phi}}$, see (2.17)). 3.3. ParameterizationWe choose a family $\Phi=\bigcup_{k=1}^{m_{\Phi}}\omega_k\subset\Pi_\Sigma$. Let $\omega=(c,c')\subset \Phi$ be one of its cells. Recall that all the cells are of the same length $\epsilon_\Phi=\tau(c,c')$. Given any $x\in\omega$, we write $x=x(r)$ if $\tau(c,x)=r$. Along with $x$, the determination set also turns out to be parameterized: $\Lambda_\Sigma[x(r)]=\{x_k(r)\}_{k=1}^{m_{\Phi}}$. When $r$ varies in the interval $(0,\epsilon_\Phi)$, the points $x_k(r)$ continuously change their position and cover the cells $\omega_k$. Thus, the family $\Phi$ is parameterized. All elements of the representations (3.3) and (3.5) are also parameterized by $r$: the vectors
$$
\begin{equation}
\begin{gathered} \, \beta^i_{\gamma\Phi}=\{(\beta^i_{\gamma\Phi})_k(r)\}_{k=1}^{m_{\Phi}}, \\ (\beta^i_{\gamma\Phi})_k(r):= \beta^i_{\gamma\Phi}(x_k(r)) =(\beta^i_{\gamma\Phi})_k=\mathrm{const},\qquad 0<r<\epsilon_\Phi, \end{gathered}
\end{equation}
\tag{3.6}
$$
and the functions $\tau^i_{\gamma\Phi}(r):=\tau^i_{\gamma\Phi}(x(r))$. By (2.20), the values of these functions are
$$
\begin{equation}
\tau^i_{\gamma\Phi}(r)=t_{i\Phi}+r \quad\text{or}\quad \tau^i_{\gamma\Phi}(r)=\widetilde{t}_{i\Phi}-r=(t_{i\Phi}+\epsilon_\Phi)-r.
\end{equation}
\tag{3.7}
$$
Note that there are two parameterizations of the family $\Phi$: one with $r=\tau(x,c)$ adopted above and the other corresponding to the parameter $r=\tau(x,c')$. They are completely equivalent. In what follows, we will assume that each family $\Phi\subset\Pi_\Sigma$ is parameterized in one of two ways. Parameterization determines matrix representations of functions and operators on the graph. Let $\Phi\subset\Pi_\Sigma$ be a parameterized family, $y\in\mathscr H$ a function on the graph, and $x=x(r)\in\Lambda_\Sigma[x(r)]=\{x_k(r)\}_{k=1}^{m_{\Phi}}\subset\Phi$, $0<r<\epsilon_{\Phi}$. We easily see that the map
$$
\begin{equation*}
U_\Phi\colon \mathscr H\to L_2([0, \epsilon_{\Phi}; \mathbb{R}^{m_{\Phi}}),\qquad (U_\Phi y)(r):= \begin{pmatrix} y(x_1(r))\\ \dots \\y(x_{m_{\Phi}}(r)) \end{pmatrix}, \quad r\in(0,\epsilon_{\Phi}),
\end{equation*}
\notag
$$
is unitary. For each vertex $\gamma\in\Sigma$ we define the following columns (which are constant; see (3.6)) and the matrices formed by them:
$$
\begin{equation*}
\beta^i_{\gamma\Phi}= \begin{pmatrix} (\beta^i_{\gamma\Phi})_1\\ \dots \\(\beta^i_{\gamma\Phi})_{m_{\Phi}} \end{pmatrix} \in \mathbb{R}^{m_{\Phi}},\qquad B_{\gamma\Phi}:= \begin{pmatrix} (\beta^1_{\gamma\Phi})_1 & \dots & (\beta^1_{\gamma\Phi})_{m_{\Phi}} \\ (\beta^{2}_{\gamma\Phi})_1 & \dots & (\beta^{2}_{\gamma\Phi})_{m_{\Phi}} \\ \dots & \dots & \dots \\ (\beta^{n_{\gamma\Phi}}_{\gamma\Phi})_1 & \dots & (\beta^{n_{\gamma\Phi}}_{\gamma\Phi})_{m_{\Phi}} \end{pmatrix}.
\end{equation*}
\notag
$$
We also introduce the matrices
$$
\begin{equation*}
D_{\gamma\Phi}(r):=\{\tau^i_{\gamma\Phi}(r)\,\delta_{ij}\}_{i,j=1}^{n_{\gamma\Phi}},\qquad r\in(0,\epsilon_{\Phi}),
\end{equation*}
\notag
$$
where $\tau^i_{\gamma\Phi}(r)$ is of the form (3.7). When the parameter $r$ varies, the matrix $B_{\gamma\Phi}$ does not change because its entries are constant on the cells $\omega_1,\dots,\omega_{m_\Phi}$ of the family $\Phi$. The matrix $B_{\gamma\Phi}^*B_{\gamma\Phi}$ is also constant. Since the columns of $\beta^i_{\gamma\Phi}$ constitute an orthonormal set, the matrix $B_{\gamma\Phi}^*B_{\gamma\Phi}$ is the projector in ${\mathbb R}^{m_{\Phi}}$ to the subspace
$$
\begin{equation*}
\mathscr{A}_{\gamma}[\Phi]:=\operatorname{span}\{\beta^1_{\gamma\Phi}, \dots, \beta^{n_{\gamma\Phi}}_{\gamma\Phi}\}=[B^*_{\gamma\Phi} B_{\gamma\Phi}] \mathbb{R}^{m_{\Phi}}.
\end{equation*}
\notag
$$
The matrix-projector $B_{\gamma\Phi}^*B_{\gamma\Phi}$ can be decomposed into a sum of pairwise orthogonal one-dimensional projectors
$$
\begin{equation}
B_{\gamma\Phi}^*B_{\gamma\Phi}=\sum_{i=1}^{n_{\gamma\Phi}}P^i_{\gamma\Phi},\qquad P^i_{\gamma\Phi}:=\langle\,{\cdot}\,,\beta^i_{\gamma\Phi}\rangle\,\beta^i_{\gamma\Phi},
\end{equation}
\tag{3.8}
$$
where $\langle\,{\cdot}\,,{\cdot}\,\rangle$ is the standard inner product in $\mathbb R^{m_\Phi}$. By (3.5), we have a representation
$$
\begin{equation*}
(U_\Phi\, E^{\,T}_\gamma\!\langle\Phi\rangle y)(r)=[B_{\gamma\Phi}^* D_{\gamma\Phi}(r) B_{\gamma\Phi}](U_\Phi{y})(r), \qquad r \in (0,\epsilon_{\Phi}),
\end{equation*}
\notag
$$
with matrices
$$
\begin{equation}
B_{\gamma\Phi}^* D_{\gamma\Phi}(r)B_{\gamma\Phi}=U_\Phi E^T_\gamma\langle\Phi\rangle U^{-1}_\Phi\stackrel{(3.5),\ (3.8)}{=} \sum_{i=1}^{n_{\gamma\Phi}}\tau_{\gamma\Phi}^i(r) P_{\gamma\Phi}^i.
\end{equation}
\tag{3.9}
$$
We now describe the parameterization of spaces and operators corresponding to the whole partition $\Pi_{\Sigma}$. The analog of the decomposition (3.1) takes the form
$$
\begin{equation*}
\mathscr{U}^T_\Sigma = \oplus \sum_{\Phi \subset \Pi_\Sigma} \mathscr{U}^T_\Sigma\langle\Phi\rangle,\qquad \mathscr{U}^T_\Sigma\langle\Phi\rangle:=\operatorname{span} \{\mathscr{U}^T_\gamma\langle\Phi\rangle\mid \gamma\in\Sigma\},
\end{equation*}
\notag
$$
and every summand reduces all eikonals simultaneously:
$$
\begin{equation*}
E^T_\gamma\mathscr{U}^T_\Sigma\langle\Phi\rangle\subset\mathscr{U}^T_\Sigma\langle\Phi\rangle, \qquad \gamma\in\Sigma.
\end{equation*}
\notag
$$
Using the parametrizations in the families, we have
$$
\begin{equation*}
U_\Phi \mathscr{H}\langle \Phi \rangle = L_2([0,\epsilon_\Phi];\mathbb{R}^{m_\Phi}),\qquad U_\Phi \mathscr{U}^T_\Sigma[\Phi] = L_2([0,\epsilon_\Phi];\mathscr{A}_\Sigma[\Phi]),
\end{equation*}
\notag
$$
where $\mathscr{A}_\Sigma[\Phi]: =\operatorname{span}\{\mathscr{A}_\gamma[\Phi]\mid \gamma\in\Sigma\}$ and the parts $E^T_\gamma\langle{\Phi}\rangle$ of the eikonals multiply the elements of $L_2([0,\epsilon_\Phi];{\mathbb R}^{m_\Phi})$ by the matrix-valued function (3.9). Convention 1. We specify our notation. The sum $\mathscr{S}=\bigoplus\sum_j \mathscr{S}_j$ of spaces $\mathscr S_1,\dots,\mathscr S_n$ is the space of $n$-tuples $s=\{s_1,\dots,s_n\}$, $s_j\in\mathscr S_j$ (with componentwise operations). The sum $A=\bigoplus\sum_j A_j\in \operatorname{End}\mathscr{S}$ of operators $A_1,\dots, A_n$, where $A_j\in\operatorname{End}\mathscr{S}_j$, is the operator acting by the rule $As:=\{A_1s_1,\dots,A_ns_n\}$. The sum $M=\bigoplus\sum_j M_j\in\mathbb M^{\varkappa_1+\dots+\varkappa_n}$ of matrices $M_1,\dots,M_n$, where $M_j\in \mathbb M^{\varkappa_j}$, is the block-diagonal matrix with blocks $M_1,\dots, M_n$ (we also write $[M]_j=M_j$). Given any algebras $\{\mathfrak{A}_1,\dots, \mathfrak{A}_n\}$, we write $\mathfrak{A}=\bigoplus_j\mathfrak{A}_j$ for their direct sum (we also write $[\mathfrak{A}]_j=\mathfrak{A}_j$ and refer to each $\mathfrak{A}_j$ as a block). The parameterization of the whole $\Pi_\Sigma$ is given by the operator $U:=\bigoplus\!\sum\limits_{\Phi\subset\Pi_\Sigma} \! U_\Phi$:
$$
\begin{equation}
\begin{aligned} \, \notag U\mathscr{H}\langle \Omega^T_\Sigma \rangle &=\bigoplus\sum_{\Phi\subset\Pi_\Sigma}L_2([0,\epsilon_\Phi];\mathbb{R}^{m_\Phi}), \\ UE^T_\gamma U^{-1} &=\bigoplus\sum_{\Phi\subset\Pi_\Sigma}U_\Phi E^T_\gamma \langle\Phi\rangle U^{-1}_\Phi \stackrel{(3.9)}{=} \bigoplus\sum_{\Phi\subset\Pi_\Sigma}\sum_{i=1}^{n_{\gamma\Phi}}\tau^i_{\gamma\Phi}P^i_{\gamma \Phi}, \qquad \gamma\in\Sigma. \end{aligned}
\end{equation}
\tag{3.10}
$$
3.4. Shifted eikonalsFor technical reasons, the following operators (shifted eikonals) are more convenient for the purpose of describing the algebra generated by eikonals:
$$
\begin{equation}
\dot E^T_{\gamma}:=\int_0^T(s+1)\,dP^s_{\gamma} = E^T_{\gamma}+ P^T_{\gamma}.
\end{equation}
\tag{3.11}
$$
The previously established properties and representations for $E^T_\gamma$ are obviously reformulated for $\dot E^T_\gamma$. Thus, the analog of the representation (3.10) takes the form
$$
\begin{equation}
\begin{aligned} \, \notag U\dot E^T_{\gamma}U^{-1} &=\bigoplus\sum_{\Phi\subset\Pi_\Sigma}U_\Phi \dot E^T_{\gamma}\langle\Phi\rangle U^{-1}_\Phi =\bigoplus\sum_{\Phi\subset\Pi_\Sigma}[B_{\gamma\Phi}^* \dot D_{\gamma\Phi}(\, {\cdot}\,) B_{\gamma\Phi}] \\ &\!\!\!\!\!\stackrel{(3.10)}{=} \bigoplus\sum_{\Phi\subset\Pi_\Sigma}\sum_{i=1}^{n_{\gamma\Phi}}\dot\tau^i_{\gamma \Phi}P^i_{\gamma \Phi}, \qquad \gamma\in\Sigma, \end{aligned}
\end{equation}
\tag{3.12}
$$
where $\dot D_{\gamma\Phi}(\,{\cdot}\,):=D_{\gamma\Phi}(\,{\cdot}\,)+ I$, $I$ is the identity matrix of appropriate dimension and $\dot\tau_{\gamma\Phi}^i(r) :=\tau_{\gamma\Phi}^{i}(r)+1$. Here is the analog of Proposition 1. Proposition 2. For the operator $\dot E^T_{\gamma}$ we have ${\operatorname{Ran}\dot E^T_{\gamma}}=\mathscr{U}^T_\gamma$, $\operatorname{Ker}\dot E^T_{\gamma}=\mathscr{H}\ominus\mathscr{U}^T_\gamma$ and $\dot E^T_{\gamma}\mathscr{U}^T_\gamma\langle\Phi\rangle\subset\mathscr{U}^T_\gamma \langle\Phi\rangle$, $\Phi\subset\Pi$. When $T<T_\gamma$, this operator has eigenvalue $0$ of infinite multiplicity and a simple absolutely continuous spectrum filling the interval $[1,T+1]$. When $T>T_\gamma$, it follows from Remark 1 that where $T_\gamma\leqslant T_0<T_1<\dots <T_N\leqslant T$, and the intervals are unions of the ranges of the functions $\dot\tau^i_{\gamma\Phi}$ (the shifts of the cells $\dot\psi^i_{\gamma\Phi}$ by $1$). § 4. The algebra of eikonals4.1. Definitions and general factsRecall that a $C^*$-algebra $\mathfrak{A}$ is a Banach algebra with an involution $a \to a^*$ satisfying the following identity [24], [25]: In particular, this is the case for the algebra $\mathfrak{B}(\mathscr{H})$ of bounded operators in a Hilbert space $\mathscr{H}$ with the operator conjugation as the involution. Writing ${\mathfrak A}\cong{\mathfrak B}$, we mean that there is an isometric $*$-isomorphism between the $C^*$-algebras ${\mathfrak A}$ and ${\mathfrak B}$ (briefly, they are isomorphic). Given a set $S\subset \mathfrak{A}$, we write $\vee S$ for the minimal $C^*$-(sub)algebra in $\mathfrak{A}$ containing $S$.By $\mathbb{M}^n$ we mean the algebra of real $(n\times n)$-matrices regarded as operators in $\mathbb{M}^n$ and endowed with the operator norm. It is irreducible. We write $C([a,b],\mathbb{M}^n)$ for the algebra of continuous $\mathbb{M}^n$-valued functions with norm $\|c\|=\sup_{a\leqslant t\leqslant b} \|c(t)\|_{\mathbb{M}^n}$. By the same symbol, we denote the operator (sub)algebra in $\mathfrak{B}(L_2([a,b];\mathbb{M}^n))$ whose elements multiply $\mathbb R^n$-valued square-summable functions by functions in $C([a,b],\mathbb{M}^n)$. The correspondence $c\mapsto c\cdot$ establishes an isomorphism between these algebras. A $C^*$-subalgebra $\mathfrak A\subset \mathbb{M}^n$ is said to be irreducible if $\mathfrak A\cong \mathbb{M}^k$, where $k\leqslant n$. Such an algebra takes a block-diagonal form In a suitable basis of $\mathbb R^n$ and consists of two blocks, one of which is $\mathbb{M}^k$ and the other (if available) is equal to zero. Here is a summary of known results.8 Proposition 3. Any $C^*$-subalgebra of $\mathbb{M}^n$ is isomorphic to the direct sum $\bigoplus_{k}\mathbb{M}^{n_k}$, where $\sum_{k}{n_k}\leqslant n$. Proposition 4 (see [24]). Suppose that $\mathfrak{P}\subset\mathbb{M}^n$ and let $\mathfrak{A}\subset C([a,b]; \mathfrak{P})$ be a $C^*$-subalgebra such that for any $t,t'\in[a,b]$ and $p,p'\in\mathfrak{P}$ there exists an element $u\in\mathfrak{A}$ with $u(t)=p$ and $u(t')=p'$. Then $\mathfrak{A}=C([a,b];\mathfrak{P})$. A representation of a $C^*$-algebra $\mathfrak{A}$ is a homomorphism $\pi\colon \mathfrak{A}\to\mathfrak{B}(H)$, where $H$ is a Hilbert space. Equivalence $\pi\sim\pi'$ of representations means that $\iota\,\pi(a)=\pi'(a)\iota$, $a\in\mathfrak{A}$, where $\iota\colon H\to H'$ is an isometry of the representation spaces. A representation is irreducible if the operators $\pi(\mathfrak{A})$ have no common non-zero invariant subspace in $H$. The spectrum of a $C^*$-algebra $\mathfrak{A}$ is the set $\widehat{\mathfrak{A}}$ of the equivalence classes of its irreducible representations. The equivalence class (the point of the spectrum) corresponding to a representation $\pi$ will be denoted by $\widehat\pi$. The spectrum is endowed with the canonical Jacobson topology [24], [25]. An isomorphism $\mathrm{u}\colon\mathfrak{A}\to\mathfrak{B}$ of $C^*$-algebras determines a correspondence between their representations,
$$
\begin{equation}
\begin{aligned} \, \widehat{\mathfrak{A}}\ni\pi\to\mathrm{u}_*\pi\in\widehat{\mathfrak{B}}, \quad(\mathrm{u}_*\pi)(b):=\pi(\mathrm{u}^{-1}(b)), \qquad b\in\mathfrak{B}, \end{aligned}
\end{equation}
\tag{4.1}
$$
which extends to a canonical homeomorphism between their spectra:
$$
\begin{equation}
\begin{aligned} \, \widehat{\mathfrak{A}}\ni\widehat\pi\to\mathrm{u}_*\widehat\pi\in\widehat{\mathfrak{B}},\qquad \mathrm{u}_*\widehat\pi:=\{\mathrm{u}_*\pi\mid \pi\in\widehat\pi\}. \end{aligned}
\end{equation}
\tag{4.2}
$$
Proposition 5. The representations are irreducible and their equivalence classes exhaust the spectrum of the algebra $C([a,b],\mathbb{M}^n)$. For every irreducible representation $\pi$ of $C([a,b],\mathbb{M}^n)$ there is a unique point $t\in[a,b]$ such that $\pi\sim\pi_t$. We put
$$
\begin{equation*}
\dot C([a,b]; \mathbb M^n):= \{\phi\in C([a,b]; \mathbb M^n)\mid \phi(a)\in \mathbb M_a,\, \phi(b)\in \mathbb M_b\},
\end{equation*}
\notag
$$
where $\mathbb M_a,\mathbb M_b$ are $C^*$-subalgebras of $\mathbb M^n$, which are referred to as boundary subalgebras. It follows from Proposition 3 that
$$
\begin{equation}
\mathbb M_a\cong\bigoplus_{k=1}^{n_a} \mathbb M^{\varkappa_k},\quad \varkappa_1+\dotsb+\varkappa_{n_a}\leqslant n;\!\!\!\! \qquad \mathbb M_b\cong\bigoplus_{k=1}^{n_b}\mathbb M^{\lambda_k},\quad \lambda_1+\dotsb+\lambda_{n_b}\leqslant n.
\end{equation}
\tag{4.4}
$$
When $\mathbb M_a=\mathbb M_b=\mathbb M^n$, we have $\dot C([a,b]; \mathbb M^n)= C([a,b]; \mathbb M^n)$. The algebras $\dot C([a,b]; \mathbb M^n)$ will be referred to as standard. The spectrum of a standard algebra consists of the classes $\widehat\pi_t$, $t\in(a,b)$, of irreducible representations of the form (4.3) and the representations $\widehat\pi_a$, $\widehat\pi_b$, which may be reducible. If, for example, $n_a\geqslant 2$, then $\pi_a$ splits into the irreducible representations
$$
\begin{equation*}
\pi_a^k\colon \phi(a)\mapsto [\phi(a)]^k\in\mathbb M^{\varkappa_k},
\end{equation*}
\notag
$$
where $[\,{\cdots}\,]^k$ is the $k$th block of the block-diagonal matrix in the representation (4.4). In this case we say that $\widehat\pi_a^1,\dots,\widehat\pi_a^{n_a}$ form a cluster in the spectrum of the standard algebra. This terminology is motivated by the fact that they are inseparable from each other in Jacobson’s topology [27]. A similar cluster may exist at the right end $t=b$. At the same time, all the points $\widehat\pi_t$ with $t\in(a,b)$ are separable from each other and from the clusters (see [7], [8]). The spectrum of $C([a,b]; \mathbb M^n)$ contains no clusters. The central object of this paper is the eikonal algebra of the graph $\Omega$,
$$
\begin{equation*}
\mathfrak{E}^T_{\Sigma}:=\vee\{E^T_{\gamma}\mid \gamma\in\Sigma\}\subset\mathfrak{B}(L_2(\Omega))
\end{equation*}
\notag
$$
(see [7], [8]). The subalgebras
$$
\begin{equation*}
\mathfrak{E}^T_{\gamma}:=\vee E^T_\gamma,\qquad \gamma\in\Sigma,
\end{equation*}
\notag
$$
are said to be partial. It is convenient to use the “shifted” algebras
$$
\begin{equation*}
\dot{\mathfrak{E}}^T_{\gamma}:=\vee \dot E^T_\gamma,\qquad\dot{\mathfrak{E}}^T_{\Sigma}:=\vee\{\dot E^T_{\gamma}\mid \gamma\in\Sigma\}= \vee\{\dot{\mathfrak{E}}^T_{\gamma}\mid \gamma\in\Sigma\}.
\end{equation*}
\notag
$$
The passage from $\mathfrak{E}^T_{\gamma}$ to $\dot{\mathfrak{E}}^T_{\gamma}$ consists in adding the projector $P^T_\gamma$, which plays the role of the identity in $\dot{\mathfrak{E}}^T_{\gamma}$ (see (3.11)) while the algebra $\mathfrak{E}^T_{\gamma}$ turns out to be a subalgebra in $\dot{\mathfrak{E}}^T_{\gamma}$. By the functional calculus of self-adjoint operators and since the projectors $P^i_{\gamma \Phi}$ in (3.12) are orthogonal, we have
$$
\begin{equation}
\varphi(\dot E^T_\gamma)=\int_{\sigma_{\mathrm{ac}}(\dot E^T_\gamma)}\varphi(s)\, dP^s_\gamma =U^{-1}\biggl[\bigoplus\sum_{\Phi\subset\Pi_\Sigma}\sum_{i=1}^{n_{\gamma\Phi}} (\varphi\circ\dot\tau^i_{\gamma\Phi})P^i_{\gamma \Phi}\biggr]U
\end{equation}
\tag{4.5}
$$
for $\varphi\in C(\sigma_{\mathrm{ac}}(\dot E^T_\gamma))$. The correspondence $\varphi(\dot E^T_\gamma)\leftrightarrow\varphi$ given by the first equality is an isomorphism of the algebras $\dot{\mathfrak{E}}^T_\gamma$ and $C(\sigma_{\mathrm{ac}}(\dot E^T_\gamma))$. Convention 2. Hereafter, unless otherwise specified, we deal only with shifted eikonals and omit $(\,\dot{}\,)$ in the notation: $\dot E^T_\gamma\equiv E^T_\gamma$, $\dot\tau^k_{\gamma l}\equiv\tau^k_{\gamma l}$, $\dot\psi^k_{\gamma l}\equiv\psi^k_{\gamma l}$, $\dot{\mathfrak{E}}^T_{\Sigma}\equiv\mathfrak{E}^T_{\Sigma}$ and so on. 4.2. Representations and connections between blocksBy (3.2) and (3.5) we have a representation
$$
\begin{equation*}
\mathfrak{E}^T_\Sigma=\vee\left\{\bigoplus\sum_{\Phi\subset\Pi_\Sigma} E_\gamma\langle\Phi\rangle\,\,\bigg|\,\,\gamma\in\Sigma\right\}.
\end{equation*}
\notag
$$
The parameterization (3.10) gives us a representation of the eikonal algebra
$$
\begin{equation}
\begin{aligned} \, \notag \mathfrak{E}^T_\Sigma &\cong U\mathfrak{E}^T_\Sigma U^{-1} \\ &\!\!\!\!\stackrel{(3.12)}{=} \vee\biggl\{\bigoplus\sum_{\Phi\subset\Pi_\Sigma}\sum_{i=1}^{n_{\gamma\Phi}} \tau^i_{\gamma\Phi}(\,{\cdot}\,) P^i_{\gamma\Phi}\biggm| \gamma\in\Sigma\biggr\}\subset\bigoplus_{\Phi\subset\Pi_\Sigma} C([0,\epsilon_\Phi],\mathbb{M}^{m_{\Phi}}) \end{aligned}
\end{equation}
\tag{4.6}
$$
in the form of an operator algebra. Its elements multiply the functions in the representation space
$$
\begin{equation}
\mathscr{R}^T_\Sigma:=\bigoplus\sum_{\Phi\subset\Pi_\Sigma} L_2([0,\epsilon_\Phi];{\mathbb R}^{m_{\Phi}})
\end{equation}
\tag{4.7}
$$
by the appropriate continuous matrix-valued functions. In the more explicit block-matrix notation for the representation (4.6) we have
$$
\begin{equation}
\begin{aligned} \, \notag U\mathfrak{E}^T_\Sigma U^{-1} &= \vee\left\{\begin{pmatrix} {\displaystyle\sum_{i=1}^{n_{\gamma \Phi^1}} \tau^i_{\gamma \Phi^1}(\cdot_1) P^i_{\gamma \Phi^1}}\\ &\ddots\\\\ &&{\displaystyle\sum_{i=1}^{n_{\gamma \Phi^J}} \tau^i_{\gamma\Phi^J}(\cdot_J) P^i_{\gamma \Phi^J}} \end{pmatrix}\Biggm| \gamma\in\Sigma\right\} \\ &\subset \begin{pmatrix} C([0,\epsilon_1]; \mathbb{M}^{m_{\Phi^1}})\\ & \ddots\\\ &&C([0,\epsilon_J]; \mathbb{M}^{m_{\Phi^J}}) \end{pmatrix}. \end{aligned}
\end{equation}
\tag{4.8}
$$
Here the zero off-diagonal blocks are omitted and $\Pi_\Sigma=\Phi^1\,\cup\,\dotsb\, \cup\,\Phi^J$. The notation $(\,\cdot_j)$ emphasizes that the arguments $r_j\in[0,\epsilon_j]$ of the functions $\tau^i_{\gamma \Phi^j}$ are different in accordance with the representation (4.7). Consider the tuples of projectors $\mathbb P_{\Phi^j}:=\{P^i_{\gamma \Phi^j}\mid i=1,\dots,n_{\gamma \Phi^j};\, \gamma\in\Sigma\}$. Using the algebras
$$
\begin{equation}
\mathfrak{P}_{\Phi^j}:=\vee\mathbb P_{\Phi^j}\subseteq\mathbb{M}^{m_{\Phi^j}},
\end{equation}
\tag{4.9}
$$
we can specify the embedding in (4.8) as follows:
$$
\begin{equation}
U\mathfrak{E}^T_\Sigma{ U^{-1}}\subset\bigoplus_{j=1}^J C([0,\epsilon_j];\mathfrak{P}_{\Phi^j}).
\end{equation}
\tag{4.10}
$$
As suggested by (4.8) and (4.10), the description of the structure of the eikonal algebra is reduced to specifying the connections between its blocks $[U\mathfrak{E}^T_\Sigma U^{-1}]_j(\,\cdot_j)$ corresponding to different families $\Phi^j\subset\Pi_\Sigma$. It is these connections that distinguish $U\mathfrak{E}^T_\Sigma U^{-1}$ from the algebra in the right-hand side of (4.10), whose blocks are completely independent. The following lemma is a step towards studying the relations between the blocks of the algebra $U\mathfrak{E}^T_\Sigma{ U^{-1}}$. We introduce the projectors
$$
\begin{equation*}
\mathcal P^i_{\gamma \Phi^j}{:=} \begin{pmatrix} O_1 \\ & \ddots\\ && P^i_{\gamma \Phi^j}\\ && \ddots\\ &&&& O_{\mathcal J} \end{pmatrix} {\in} \begin{pmatrix} \mathfrak{P}_{\Phi^1} \\ & \ddots\\\ && \mathfrak{P}_{\Phi^j}\\ && \ddots\\ &&&& \mathfrak{P}_{\Phi^J} \end{pmatrix} {=}\bigoplus_{j=1}^J \mathfrak{P}_{\Phi^j},
\end{equation*}
\notag
$$
where $O_k$ are the zero matrices of appropriate dimensions. Like $P^i_{\gamma \Phi^j}$, these projectors are pairwise orthogonal. We also define the “points” $\mathbf{r}:=\{r_1,\dots,r_J\}$ with coordinates $r_j\in[0,\epsilon_j]$, the matrices
$$
\begin{equation}
(U{ E}^T_\gamma{ U^{-1}})(\mathbf{r}) : =\bigoplus\sum_{j=1}^J\sum_{i=1}^{n_{\gamma\Phi^j}}\tau^i_{\gamma \Phi^j}(r_j) P^i_{\gamma\Phi^j}=\sum_{j=1}^J\sum_{i=1}^{n_{\gamma\Phi^j}}\tau^i_{\gamma\Phi^j}(r_j) \mathcal P^i_{\gamma\Phi^j} \in\bigoplus_{j=1}^J \mathfrak{P}_{\Phi^j}
\end{equation}
\tag{4.11}
$$
(see (3.12)) and the matrix algebras
$$
\begin{equation*}
(U\mathfrak{E}^T_\Sigma U^{-1})(\mathbf{r}):=\vee \{(UE^T_\gamma U^{-1})(\mathbf{r})\mid \gamma\in\Sigma\}.
\end{equation*}
\notag
$$
Lemma 2. Fix $\gamma$, $i$, $j$. Let the points $\mathbf{r}$ and $\mathbf{r}'$ be such that their coordinates satisfy $r_j\in(0,\epsilon_j)$ and ${r_j}\ne {r_j'}$. Then there exists an element $e\in U\mathfrak{E}^T_\Sigma U^{-1}$ such that $e(\mathbf{r})=\mathcal{P}^i_{\gamma j}$ and $e(\mathbf{r}')=O$. Proof. Since the projectors in (4.11) are pairwise orthogonal, for $s\in\mathbb N$ we have Hence the following equality holds for the polynomial $q=q(t)=a_\nu t^\nu+\dots+a_1t$:
By the condition on the coordinates of $\mathbf{r}$ and $\mathbf{r}'$ and the property (2.19), the number $\tau^i_{\gamma j}(r_j)$ occurs once in the set of numbers Choose the polynomial in such a way that it equals $1$ at $t=\tau^i_{\gamma j}(r_j)$ and $0$ at all other points in this set. Putting $e:=q(UE^T_\gamma U^{-1})$, we obviously have $e(\mathbf{r})= \mathcal P^i_{\gamma j}$, $e(\mathbf{r}')=O$. $\Box$Corollary 2. If the coordinates of the point $\mathbf{r}$ do not take extreme values, i.e. $r_j\notin\{0,\epsilon_j\}$ for all $j$, then If the coordinates are such that $0<a_j\leqslant r_j\leqslant b_j<\epsilon_j$ for a fixed $j$, and the other coordinates are arbitrary, then The first relation follows directly from the lemma, and the second can easily be derived from the first by using Proposition 4. By (4.13), there are no connections between the blocks $[U\mathfrak{E}^T_\Sigma U^{-1}]_j$ under these constraints on the coordinates. If the coordinates of $\mathbf{r}$ take extreme values, (4.12) does not hold: there can be links between the blocks in the matrix algebra $[U\mathfrak{E}^T_\Sigma U^{-1}](\mathbf{r})$. It is these links that distinguish the left- and right-hand sides in the embedding (4.10): the latter consists of standard algebras and has no such links. We explain this by examples. It follows from the definition and properties of the functions $\tau^i_{\gamma \Phi^j}$ (see (2.19)) that the equality $\tau_{\gamma \Phi^j}^i(r_j)=\tau_{\gamma\Phi^{j'}}^{i'}(r_{j'})$ is possible only at extreme values of the coordinates, i.e. when $r_j\in\{0,\epsilon_j\}$ and $r_{j'}\in\{0,\epsilon_{j'}\}$. Let the functions $\tau^i_{\gamma\Phi^j}$ be such that $\tau^i_{\gamma\Phi^j}(\epsilon_j)=\tau^{i+1}_{\gamma\Phi^j}(\epsilon_j)=\tau$. Then the $j$th block of the eikonal in (4.8) is of the form
$$
\begin{equation*}
\bigl[[UE^T_\gamma U^{- 1}](\mathbf{r})\bigr]_j=\tau^1_{\gamma\Phi^j}(\epsilon_j) P^1_{\gamma \Phi^j}+\dots+\tau( P^i_{\gamma\Phi^j}+ P^{i+1}_{\gamma \Phi^j}) +\dots+\tau^{n_{\gamma\Phi^j}}_{\gamma \Phi^j}(\epsilon_j)P^{n_{\gamma\Phi^j}}_{\gamma \Phi^j}.
\end{equation*}
\notag
$$
Hence the projectors $P^i_{\gamma \Phi^j}$ and $P^{i+1}_{\gamma \Phi^j}$ do not occur separately in the family of generators of the algebra $[U\mathfrak{E}^T_\Sigma U^{-1}](\mathbf{r})$, but only their sum $P^i_{\gamma \Phi^j}+ P^{i+1}_{\gamma\Phi^j}$ occurs and, therefore, the number of generators decreases by $1$. As a consequence, (4.12) need not hold (and indeed does not hold in interesting cases): instead of equality, we can guarantee only an inclusion $[U\mathfrak{E}^T_\Sigma U^{-1}](\mathbf{r})\subset\bigoplus_{j=1}^J \mathfrak{P}_{\Phi^j}$. Projectors belonging to distinct blocks may also be linked. Let $\tau^i_{\gamma \Phi^j}$ be such that $\tau^i_{\gamma\Phi^j}(\epsilon_j)=\tau^{i'}_{\gamma\Phi^{j'}}(\epsilon_{j'})=\tau$ for distinct $j$ and $j'$. Then the blocks with numbers $j$ and $j'$ in (4.8) are of the form
$$
\begin{equation*}
\begin{aligned} \, \bigl[[UE^T_\gamma U^{- 1}](\mathbf{r})\bigr]_j &=\tau^1_{\gamma\Phi^j}(\epsilon_j) P^1_{\gamma \Phi^j}+\dots+\tau P^i_{\gamma\Phi^j} +\dots+\tau^{n_{\gamma\Phi^j}}_{\gamma \Phi^j}(\epsilon_j) P^{n_{\gamma\Phi^j}}_{\gamma \Phi^j}, \\ \bigl[[U\, E^T_\gamma U^{-1}](\mathbf{r})\bigr]_{j'} &=\tau^1_{\gamma \Phi^{j'}}(\epsilon_{j'})P^1_{\gamma \Phi^{j'}}+\dots+\tau P^{i'}_{\gamma \Phi^{j'}} +\dots+\tau^{n_{\gamma\Phi^{j'}}}_{\gamma\Phi^{j'}}(\epsilon_{j'}) P^{n_{\gamma\Phi^{j'}}}_{\gamma\Phi^{j'}}. \end{aligned}
\end{equation*}
\notag
$$
Hence they are related (by means of the projectors $P^i_{\gamma\Phi^j}$ and $P^{i'}_{\gamma \Phi^{j'}}$). This may also result in a decrease of the number of generators of the algebra $[U\mathfrak{E}^T_\Sigma U^{-1}](\mathbf{r})$. 4.3. ReducibilityGenerally speaking, the algebras $\mathfrak{P}_{\Phi^j}$ defined by the families of projectors $\mathbb P_{\Phi^j}$ in (4.9) are reducible. By Proposition 3, we have
$$
\begin{equation*}
\mathfrak{P}_{\Phi^j} = \bigoplus_{k=1}^{q_j}\mathfrak{P}_{\Phi^j}^k,\qquad \mathfrak{P}_{\Phi^j}^k \cong \mathbb M^{\varkappa_{j,k}}, \quad \varkappa_{j,1}+\dots+ \varkappa_{j,q_k}\leqslant m_{\Phi^j},
\end{equation*}
\notag
$$
and the embedding (4.10) takes the form
$$
\begin{equation}
U\mathfrak{E}^T_\Sigma{ U^{-1}}\subset\bigoplus_{j=1}^J C\biggl([0,\epsilon_j];\bigoplus_{k=1}^{q_j}\mathfrak{P}_{\Phi^j}^k\biggr)= \bigoplus_{j=1}^J\bigoplus_{k=1}^{q_j}C([0,\epsilon_j];\mathfrak{P}_{\Phi^j}^k)
\end{equation}
\tag{4.14}
$$
with irreducible $\mathfrak{P}_{\Phi^j}^k$. In algebraic Theorem 1 below, among other results, we show that
$$
\begin{equation}
\mathfrak{P}_{\Phi^j}^k = \vee \mathbb P_{\Phi^j}^k,
\end{equation}
\tag{4.15}
$$
where $\mathbb P_{\Phi^j}^k$ is a subset of $\mathbb P_{\Phi^j}$ such that
$$
\begin{equation}
\bigcup_{k=1}^{q_j} \mathbb P_{\Phi^j}^k=\mathbb P_{\Phi^j},\qquad \mathbb P_{\Phi^j}^k \cap \mathbb P_{\\Phi^j}^{k'}=\varnothing\quad\text{for} \quad k\neq k'.
\end{equation}
\tag{4.16}
$$
Thus, writing the algebra $U\mathfrak{E}^T_\Sigma{ U^{-1}}$ as a sum of irreducible blocks in (4.14) amounts to an adequate grouping of projectors within every set $\mathbb P_{\Phi^j}$. When grouping, it is convenient to relabel the projectors. The first step is to successively relabel the algebras occurring in (4.14):
$$
\begin{equation}
\begin{aligned} \, \notag &\mathfrak{P}_{\Phi^1}^1,\, \dots,\, \mathfrak{P}_{\Phi^1}^{q_1};\quad \mathfrak{P}_{\Phi^2}^1,\, \dots,\, \mathfrak{P}_{\Phi^2}^{q_2}; \quad\dots;\quad \mathfrak{P}_{\Phi^J}^1,\, \dots,\, \mathfrak{P}_{\Phi^J}^{q_J} \\ &\qquad\Longrightarrow \mathfrak{P}_1,\, \dots,\, \mathfrak{P}_{q_1};\quad \mathfrak{P}_{q_1+1},\, \dots,\, \mathfrak{P}_{q_1+q_2}; \quad \dots;\quad \mathfrak{P}_{q_1+s+q_{J-1}},\, \dots,\, \mathfrak{P}_L, \end{aligned}
\end{equation}
\tag{4.17}
$$
where $L:={q_1+\dots+q_{J}}$. Formally replacing $\mathfrak{P}$ by $\mathbb P$ in (4.17), we similarly pass from the sets $\mathbb P_{\Phi^j}^k$ to the new sets $\mathbb P_{l}$, where $l=1,\dots, L$. In its turn, each $\mathbb P_{l}$ splits into the subsets corresponding to the individual vertices $\gamma\in\Sigma$:
$$
\begin{equation*}
\mathbb P_l=\bigcup_{\gamma\in\Sigma}\mathbb P_{l}^{\gamma},\qquad \mathbb P_{l}^{\gamma} =\{P^i_{\gamma' \Phi^j}\in\mathbb P_{l}\mid \gamma'=\gamma\}=\{P^{i_1}_{\gamma\Phi^j},\dots,P^{i_{n_{\gamma l}}}_{\gamma \Phi^j}\},
\end{equation*}
\notag
$$
where $n_{\gamma l}:=\#\mathbb P_{l}^{\gamma}$. Finally, we renumber the projectors inside each $\mathbb P_{l}^{\gamma}$:
$$
\begin{equation*}
P^{i_1}_{\gamma \Phi^j},\ \dots,\ P^{i_k}_{\gamma \Phi^j},\ \dots, \ P^{i_{n_{\gamma l}}}_{\gamma \Phi^j}\quad \Longrightarrow\quad P^1_{\gamma l},\ \dots,\ P^{k}_{\gamma l},\ \dots,\ P^{n_{\gamma l}}_{\gamma l}
\end{equation*}
\notag
$$
as well as the functions $\tau_{\gamma \Phi^j}^i$ and the projectors $\mathcal P_{\gamma \Phi^j}^i$:
$$
\begin{equation*}
\begin{aligned} \, &\tau^{i_1}_{\gamma \Phi^j},\, \dots,\, \tau^{i_k}_{\gamma \Phi^j},\,\dots,\, \tau^{i_{n_{\gamma l}}}_{\gamma \Phi^j}; \quad \mathcal P^{i_1}_{\gamma \Phi^j},\, \dots,\, \mathcal P^{i_k}_{\gamma \Phi^j},\, \dots,\, \mathcal P^{i_{n_{\gamma l}}}_{\gamma \Phi^j} \\ &\qquad\Longrightarrow \tau^1_{\gamma l},\, \dots,\, \tau^{k}_{\gamma l},\, \dots,\, \tau^{{n_{\gamma l}}}_{\gamma l}; \quad \mathcal P^1_{\gamma l},\, \dots,\, \mathcal P^{k}_{\gamma l},\, \dots,\, \mathcal P^{n_{{\gamma l}}}_{\gamma l}. \end{aligned}
\end{equation*}
\notag
$$
The embedding (4.14) can be written in the new notation as
$$
\begin{equation}
U\mathfrak{E}^T_\Sigma{ U^{-1}}\subset\bigoplus_{l=1}^{L} C([0,\varepsilon_l];\mathfrak{P}_{l}),
\end{equation}
\tag{4.18}
$$
where $\varepsilon_1=\dots =\varepsilon_{q_1}=\epsilon_1$; $\varepsilon_{q_1+1}=\dots =\varepsilon_{q_1+q_2}=\epsilon_2$; $\dots$ . The eikonal representation (3.10) and the relation (4.12) take the following form consistent with (4.18):
$$
\begin{equation}
UE^T_\gamma U^{- 1}= \bigoplus\sum_{l=1}^L \biggl[\sum_{k=1}^{n_{\gamma l}}\tau^k_{\gamma l} P^k_{\gamma l}\biggr] =\sum_{l=1}^L \biggl[\sum_{k=1}^{n_{\gamma l}}\tau^k_{\gamma l} \mathcal P^k_{\gamma l}\biggr]
\end{equation}
\tag{4.19}
$$
and
$$
\begin{equation*}
(U\mathfrak{E}^T_\Sigma U^{-1})(\mathbf{r})=\bigoplus_{l=1}^L \mathfrak{P}_l,
\end{equation*}
\notag
$$
where the coordinates of the point $\mathbf{r} = \{r_1,\dots,r_{L}\}$ do not take extreme values. This is the point where the subscript $\gamma$ of the functions $\tau^i_\Phi$ becomes necessary (see (2.26)) because the originally equal functions $\tau_{\gamma \Phi^j}^i=\tau_{\gamma'\Phi^j}^i$ for a fixed family $\Phi^j$ can correspond to the projectors $P_{\gamma \Phi^j}^i$ and $P_{\gamma'\Phi^j}^i$ which, after renumbering, belong to different blocks $\mathfrak{P}_l$ and $\mathfrak{P}_{l'}$. By Corollary 2 and the subsequent comments, connections between the blocks of the algebra $U\mathfrak{E}^T_\Sigma{U^{-1}}$ can occur only on the boundaries of the intervals $[0,\varepsilon_j]$. To describe these connections in detail, we shall use the following formalism. Fix a vertex $\gamma\in\Sigma$ and consider a triple $(k,l,r_l)$ corresponding to the value $\tau_{\gamma l}^k(r_l)$ of the function $\tau_{\gamma l}^k$. We say that triples $(k,l,r_l)$ and $(k',l',r_{l'})$ are related and write $(k,l,r_l)\leftrightarrow (k',l',r_{l'})$ if $\tau_{\gamma l}^k(r_l)=\tau_{\gamma l'}^{k'}(r_{l'})$. By the properties (2.19) and (2.20), this is possible only for extreme values of the parameters $r_l\in\{0,\varepsilon_l\}$ and $r_{l'}\in\{0,\varepsilon_{l'}\}$; only these values occur in the proposition below. The same properties lead to the following properties of the relation $\leftrightarrow$. Proposition 6. For every given triple $(k,l,r_l)$, exactly one of the following conditions holds. We divide the set of all triples $(k,l,r_l)$ into types $\mathbf 1$, $\mathbf 2$, and $\mathbf 3$ in accordance with Proposition 6. Lemma 3. Suppose that $\mathbf{r}=\{r_1,\dots, r_l,\dots,r_\mathcal L\}$, $r_l\in\{0,\varepsilon_l\}$. 1. If $(k,l,r_l)\in \mathbf 1$, then for every $\widetilde{\mathbf{r}}$ with $\widetilde r_{l}\neq r_l$ there exists an element $e\in U\mathfrak{E}^T_\Sigma U^{-1}$ such that
$$
\begin{equation*}
e(\mathbf{r})={\mathcal P}^k_{\gamma l},\qquad e(\widetilde{\mathbf{r}})=O.
\end{equation*}
\notag
$$
2. If $(k,l,r_l)\in \mathbf{2}$ and $(k,l,r_l)\leftrightarrow(k',l,r_l)$, then for every $\widetilde{\mathbf{r}}$ with $\widetilde r_{l}\neq r_l$ there exists an element $e\in U\mathfrak{E}^T_\Sigma U^{-1}$ such that
$$
\begin{equation*}
e(\mathbf{r})={\mathcal P}^k_{\gamma l}+{\mathcal P}^{k'}_{\gamma l},\qquad e(\widetilde{\mathbf{r}})=O.
\end{equation*}
\notag
$$
3. If $(k,l,r_l)\,{\in}\, \mathbf 3$ and $(k,l,r_l)\,{\leftrightarrow}\,(k',l',r_{l'})$, then for $\mathbf{r}\,{=}\,\{r_1,\dots, r_l,\dots,r_{l'},\dots,r_\mathcal L\}$ and for every $\widetilde{\mathbf{r}}$ with $\widetilde r_{j}\neq r_j$ and $\widetilde r_{j'}\neq r_{j'}$ there exists an element $e\in U\mathfrak{E}^T_\Sigma U^{-1}$ such that Proof. The proof is similar to that of Lemma 2. It consists in finding an appropriate polynomial $q$. The projectors can “glue” into sums for the same reason as in the examples after Corollary 2. $\Box$ We have already noticed that the difference between the algebras $U\mathfrak{E}^T_\Sigma U^{-1}$ and $\bigoplus_{l=1}^L \!\! C([0,\varepsilon_l]; \mathfrak{P}_l)$ consists in the possible relations between the blocks $(U\mathfrak{E}^T_\Sigma U^{-1})(\mathbf{r})$ of the matrix algebra in the case when the coordinates $r_l$ take extreme values. To study these relations, we use the boundary algebra
$$
\begin{equation}
\partial (U\mathfrak{E}^T_\Sigma U^{-1}):=\bigl\{ e(\mathbf{0})\oplus e(\boldsymbol{\varepsilon})\bigm| e\in U\mathfrak{E}^T_\Sigma U^{-1}\bigr\} \subset \biggl[ \bigoplus_{l=1}^L \mathfrak{P}_l\biggr]\oplus\biggl[ \bigoplus_{l=1}^L \mathfrak{P}_l\biggr],
\end{equation}
\tag{4.20}
$$
where $\mathbf{0}=\{0,\dots,0\}$ and $\boldsymbol{\varepsilon}=\{\varepsilon_1,\dots,\varepsilon_L\}$. Our description of these relations begins with some general results on the structure of matrix algebras of the form $\bigoplus_{l=1}^{L^{\vphantom{I}}} \mathfrak{P}_l$. 4.4. Algebras generated by one-dimensional projectorsLet $\mathscr{G}$ be a Hilbert space with inner product $\langle\,{\cdot}\,,{\cdot}\,\rangle$, and let $P^1,\dots,P^n$ be one-dimensional projectors: $P^i=\langle\,{\cdot}\,,\beta^i\rangle\beta^i$, $\|\beta^i\|=1$. We put
$$
\begin{equation*}
\mathscr A:=\operatorname{span}\{\beta^1,\dots,\beta^n\},\qquad \mathfrak P:=\vee\{P^1,\dots,P^n\}\subset\mathfrak{B}(\mathscr{G}).
\end{equation*}
\notag
$$
The set $B:=\{\beta^1,\dots,\beta^n\}$ can be endowed with a reflexive and symmetric relation by putting $\beta^i\sim_0\beta^{i'}$ if $\langle\beta^i,\beta^{i'}\rangle\neq 0$. It determines an equivalence: we have $\beta^i\sim\beta^{i'}$ if there are vectors $\beta^{i_1},\dots,\beta^{i_k}$ such that $\beta^i \sim_0\beta^{i_1}\sim_0 \cdots\sim_0\beta^{i_k}\sim_0 \beta^{i'}$. Then we transfer this equivalence to the projectors by declaring that $P^i\sim P^{i'}$ if $\beta^i\sim\beta^{i'}$. Write $B=B_1\,{\cup}\,{\cdots}\,{\cup}\, B_q$ as the union of equivalence classes and put $\mathscr A_k:=\operatorname{span}B_k$. It follows from the definition of $\sim$ that $\mathscr A_k\perp \mathscr A_l$ for $k\neq l$. Thus we have a decomposition $\mathscr A=\mathscr A_1\oplus\dots\oplus \mathscr A_q$, which obviously reduces all projectors $P^i$. Proposition 7. The algebra $\mathfrak P$ is reduced by the subspaces $\mathscr A_k$ and we have where $\mathfrak P|_{\mathscr{A}_k}\cong\mathbb{M}^{\varkappa_k}$, $\varkappa_k=\operatorname{dim}\mathscr A_k$. Reducibility is obvious; the equality for the dimensions follows since every vector $\beta^i\in\mathscr A_k$ is cyclic in $\mathscr A_k$ for the part $\mathfrak{P}|_{\mathscr A_k}$. The decomposition (4.21) can be obtained by using the procedure (2.11), (2.12). The following considerations model the situation arising in the study of the eikonal algebra. Namely, the possible relations between the blocks in the representation (4.8) will be discussed in an abstract form. Let $\mathscr{G}_k$, $k=1,2,3$, be three Hilbert spaces, each endowed with a set $P^1_k,\dots,P^{n_k}_k$ of one-dimensional projectors: $P^i_k=\langle\,{\cdot}\,,\beta^i_k\rangle\beta^i_k$, $\|\beta^i_k\|=1$, where the vectors $\beta^i_k$ belong to the sets
$$
\begin{equation*}
B_k:=\{\beta^i_k\mid i=1,\dots,{n}_k\}\subset \mathscr{G}_k.
\end{equation*}
\notag
$$
These projectors generate the algebras
$$
\begin{equation*}
\mathfrak{P}_1=\vee\{P^1_1,\dots,P_1^{n_1}\},\qquad \mathfrak{P}_2=\vee\{P_2^1,\dots,P_2^{n_2}\},\qquad \mathfrak{P}_3=\vee\{P_3^1,\dots,P_3^{n_3}\}.
\end{equation*}
\notag
$$
We construct an algebra
$$
\begin{equation}
\mathfrak{P} := \mathfrak{P}_1\oplus\mathfrak{P}_2\oplus\mathfrak{P}_3 \subset\mathfrak{B}(\mathscr{G}_1\oplus \mathscr{G}_2\oplus\mathscr{G}_3).
\end{equation}
\tag{4.22}
$$
It is a subalgebra of the algebra of bounded operators in the space $\mathscr{G}_1\oplus\mathscr{G}_2\oplus\mathscr{G}_3$, with generators
$$
\begin{equation}
\begin{gathered} \, \mathcal{P}_1^i:=P_1^i \oplus O_2 \oplus O_3,\qquad i=1,\dots,n_1, \\ \mathcal{P}_2^i:= O_1\oplus P^i_2\oplus O_3,\qquad i=1,\dots,n_2, \\ \mathcal{P}_3^i:= O_1\oplus O_2 \oplus P^i_3,\qquad i=1,\dots,n_3, \end{gathered}
\end{equation}
\tag{4.23}
$$
where $O_k$ is the zero operator acting in the $k$th component of the space $\mathscr{G}_1\oplus\mathscr{G}_2\oplus\mathscr{G}_3$. The algebras $\mathfrak{P}_k$ in (4.22) will be referred to as blocks of the algebra $\mathfrak{P}$. Note that in the current considerations, roughly speaking, the algebras $\mathfrak{P}_1$ and $\mathfrak{P}_2$ correspond to some pair of selected blocks in (4.8) while $\mathfrak{P}_3$ is “all the rest”. We say that an algebra $\mathfrak{Q}\subset\mathfrak{P}$ separates (or does not connect) the blocks $\mathfrak{P}_1$ and $\mathfrak{P}_2$ in (4.22) if, along with any element $q_1\oplus q_2\oplus q_3\in\mathfrak{Q}$, it contains the $q_1\oplus O_2\oplus q_3'$ and $O_1\oplus q_2\oplus q_3''$, where $q_3'$, $q_3''$ are some elements of $\mathfrak{P}_3$. Otherwise we say that these blocks are connected. We similarly define a connection (or its lack) for any pair of blocks in (4.22). Note the following obvious fact. If $\mathfrak{Q}$ admits a system of generators each of which is of the form either $q_1\oplus O_2\oplus q_3'$ or $O_1\oplus q_2\oplus q_3''$, then it separates $\mathfrak{P}_1$ and $\mathfrak{P}_2$. Let
$$
\begin{equation}
\mathbb P:=\{\mathcal{P}_k^i\mid i=1,\dots,{n}_k;\, k=1,2,3\}
\end{equation}
\tag{4.24}
$$
be the total set of generators of the algebra $\mathfrak{P}$. Let $\mathcal{T}\colon \mathbb P\to\mathbb P$ be a map (an involution) on $\mathbb P$ such that if $\mathcal{T}(\mathcal{P}_k^i)=\mathcal{P}_{k'}^{i'}$, then $\mathcal{T}(\mathcal{P}_{k'}^{i'})=\mathcal{P}_{k}^i$ and exactly one of the following conditions holds:
$$
\begin{equation*}
\mathcal{T}(\mathcal{P}_k^i)=\mathcal{P}_k^i\quad\text{or}\quad \mathcal{T}(\mathcal{P}_k^i)\mathcal{P}_k^i=\mathcal{P}_k^i\mathcal{T}(\mathcal{P}_k^i)=0.
\end{equation*}
\notag
$$
This map induces a partition of the set $\mathbb{P}$ into pairs $\{\mathcal P,\mathcal T(\mathcal P)\}$ such that the components of each pair are either equal or orthogonal to each other. We see that many such maps $\mathcal{T}$ exist. However, the map $\mathcal{T}$ in the eikonal algebra will be quite concrete and determined by the values of the functions $\tau^k_{\gamma l}(r)$ at $r=0$ and $r=\varepsilon$. Using this map, we define the projectors
$$
\begin{equation*}
\mathcal{Q}_k^i := \begin{cases} \mathcal{P}_k^i &\text{if }\mathcal{T}(\mathcal{P}_k^i)=\mathcal{P}_k^i, \\ \mathcal{P}_k^i+\mathcal{T}(\mathcal{P}_k^i) &\text{if } \mathcal{T}(\mathcal{P}_k^i)\mathcal{P}_k^i=0, \end{cases}
\end{equation*}
\notag
$$
some of which may coincide. If $\mathcal{T}$ is not the identity, then their total number is clearly less than $n_1+n_2+n_3$ because of coincidences. We consider an algebra
$$
\begin{equation}
\mathfrak{Q}:=\vee\{\mathcal{Q}_k^i\mid i=1,\dots,{n}_k;\,k=1,2,3\}\subset \mathfrak{P}.
\end{equation}
\tag{4.25}
$$
It is determined by the map $\mathcal T$. However, different admissible maps $\mathcal T$ and $\mathcal T'$ may yield the same algebra $\mathfrak{Q}$. This will be used in the proof of Theorem 1. We now discuss the conditions under which this algebra $\mathfrak{Q}$ separates or connects the blocks $\mathfrak{P}_1$ and $\mathfrak{P}_2$. Divide the set $\mathbb{P}$ (defined in (4.24)) into the parts $\mathbb{P}_k:=\{\mathcal{P}_k^i\mid i=1,\dots,{n}_k\}$, $k=1,2,3$. For every part we have the matrices
$$
\begin{equation*}
G(\mathbb P_k):=\{\|\mathcal{P}^i_k\mathcal{P}^j_k\|\}_{i,j=1}^{n_k}= \begin{pmatrix} \|\mathcal{P}^1_k\mathcal{P}^1_k\| &\cdots &\|\mathcal{P}^1_k\mathcal{P}^{n_k}_k\| \\ \vdots & \ddots &\vdots\\ \|\mathcal{P}^{n_k}_k\mathcal{P}^1_k\| & \cdots & \|\mathcal{P}^{n_k}_k\mathcal{P}^{n_k}_k\| \end{pmatrix}
\end{equation*}
\notag
$$
and ${G}(\mathcal{T}(\mathbb{P}_k))=\{\|\mathcal T(\mathcal{P}^i_k)\mathcal T(\mathcal{P}^j_k)\|\}_{i,j=1}^{n_k}$. Put
$$
\begin{equation*}
B:=\{\beta^i_k\mid i=1,\dots,{n}_k;\,k=1,2,3\} = B_1\cup B_2\cup B_3,\qquad B_k=\{\beta^i_k\mid i=1,\dots,{n}_k\}.
\end{equation*}
\notag
$$
By the obvious equalities $\|\mathcal{P}^i_k\,\mathcal{P}^j_{k}\|=|\langle\beta^i_k,\beta^j_{k}\rangle|$, we have
$$
\begin{equation*}
G(\mathbb{P}_k)= \begin{pmatrix} |\langle\beta_k^1,\beta_k^1\rangle| &\cdots &||\langle\beta_k^1,\beta_k^{{n}_k}\rangle| \\ \vdots & \ddots &\vdots\\ |\langle\beta_k^{{n}_k},\beta_k^1\rangle| & \cdots & |\langle\beta_k^{{n}_k},\beta_k^{{n}_k}\rangle| \end{pmatrix}.
\end{equation*}
\notag
$$
The following result on the connection of blocks will be used in our study of the structure of the eikonal algebra. Theorem 1. Let the blocks $\mathfrak{P}_1$ and $\mathfrak{P}_2$ of the algebra (4.22) be such that the corresponding sets $B_1$ and $B_2$ are equivalence classes with respect to the relation $\sim$. Then the algebra $\mathfrak{Q}$ can connect $\mathfrak{P}_1$ and $\mathfrak{P}_2$ only when Proof. 1. Given any $k,k'=1,2,3$, $k'\neq k$, we put
$$
\begin{equation*}
\mathbb{Q}_{k k'}:= \{\mathcal{Q}_k^i=\mathcal{P}_k^i+\mathcal{T}(\mathcal{P}_k^i)\mid \mathcal{T}(\mathcal{P}_k^i)\in\mathbb P_{k'}\}.
\end{equation*}
\notag
$$
Since $\mathcal T$ is an involution, we have an equality
$$
\begin{equation*}
\mathbb{Q}_{k' k} = \{\mathcal{Q}_{k'}^{i'}=\mathcal{P}_{k'}^{i'}+\mathcal{T}(\mathcal{P}_{k'}^{i'})\mid \mathcal{T}(\mathcal{P}_{k'}^{i'})\in\mathbb P_{k}\} = \mathbb{Q}_{k k'}.
\end{equation*}
\notag
$$
When $k=k'=1,2,3$, we put
$$
\begin{equation*}
\mathbb{Q}_{k k}:= \Biggl\{\mathcal{Q}_k^i= \begin{cases} \mathcal{P}_k^i+\mathcal{T}(\mathcal{P}_k^i) &\text{if }\mathcal{P}_k^i\ne \mathcal{T}(\mathcal{P}_k^i)\in\mathbb P_{k}, \\ \mathcal{P}_k^i &\text{if }\mathcal{P}_k^i=\mathcal{T}(\mathcal{P}_k^i), \end{cases}\Biggm| i=1,\dots,n_k\Biggr\}\Biggr\}.
\end{equation*}
\notag
$$
Thus, all the sets $\mathbb{Q}_{k k'}$ consist of one- and two-dimensional projectors, and the algebra $\mathfrak{Q}$ can be represented in the form
$$
\begin{equation}
\mathfrak{Q}=\vee[\mathbb{Q}_{11}\cup\mathbb{Q}_{22}\cup\mathbb{Q}_{33}\cup\mathbb{Q}_{12} \cup\mathbb{Q}_{13}\cup\mathbb{Q}_{23}].
\end{equation}
\tag{4.26}
$$
We easily see from the structure of the projectors (4.23) that the algebra $\mathfrak{Q}$ can connect $\mathfrak{P}_1$ and $\mathfrak{P}_2$ only when $\mathbb{Q}_{12}\neq\varnothing$. Otherwise it separates them.
2. Suppose that the algebra $\mathfrak{Q}$ admits a representation
$$
\begin{equation}
\mathfrak{Q}=\vee [\mathbb{Q}_{11}'\cup\mathbb{Q}_{22}'\cup\mathbb{Q}_{33}\cup\mathbb{Q}_{13}\cup \mathbb{Q}_{23}],
\end{equation}
\tag{4.27}
$$
where
$$
\begin{equation*}
\mathbb{Q}_{11}':=\mathbb{Q}_{11}\cup\,\{\mathcal{P}_1^i\mid \mathcal{P}_1^i+\mathcal T(\mathcal{P}_1^i)\in\mathbb{Q}_{12}\},\qquad \mathbb{Q}_{22}': =\mathbb{Q}_{22}\cup\,\{\mathcal{P}_2^i\mid \mathcal{P}_2^i+\mathcal T(\mathcal{P}_2^i)\in\mathbb{Q}_{12}\}.
\end{equation*}
\notag
$$
Then it separates $\mathfrak{P}_1$ and $\mathfrak{P}_2$. Indeed, in this case, along with the elements $\mathcal Q^i_1=\mathcal{P}_1^i+\mathcal T(\mathcal{P}_1^i)$ and $\mathcal Q^i_2=\mathcal{P}_2^i+\mathcal T(\mathcal{P}_2^i)$, the algebra $\mathfrak{Q}$ contains all the projectors $\mathcal{P}_1^i={P}^i_1\oplus O_2\oplus O_3$ and $\mathcal{P}_2^i=O_1\oplus{P}^i_2\oplus O_3$. By adding them to the set of generators of ${\mathfrak{Q}}$ instead of the elements of $\mathbb Q_{12}$, we easily see that separation takes place.
Here is an explanation. The definition of $\mathcal T$ excludes simultaneous presence of the elements $\mathcal Q^i_k=\mathcal P^i_k+\mathcal T(\mathcal P^i_k)$ and $\mathcal P^i_k$ in the list $\{\mathcal{Q}_k^i\mid i=1,\dots,{n}_k; \,k=1,2,3\}$ of generators of the algebra $\mathfrak Q$ (see (4.25)). Nevertheless, they may all belong to the algebra. This corresponds to the possibility of replacing $\mathcal T$ by another map $\mathcal T'\colon\mathbb{P}\to\mathbb P$ such that the corresponding algebra $\mathfrak Q'$ coincides with $\mathfrak Q$. 3. We claim that if
$$
\begin{equation*}
\mathbb{Q}_{11}\cup\mathbb{Q}_{13}\cup\mathbb{Q}_{22}\cup\mathbb{Q}_{23}\neq \varnothing,
\end{equation*}
\notag
$$
then the algebra $\mathfrak{Q}$ can be written in the form (4.27) and, therefore, it separates $\mathfrak{P}_1$ and $\mathfrak{P}_2$. Indeed, since $\mathfrak{P}_1$ and $\mathfrak{P}_2$ play the same role, our considerations for the parts $\mathbb{Q}_{11}\cup\mathbb{Q}_{13}$ and $\mathbb{Q}_{22}\cup\mathbb{Q}_{23}$ will be similar to each other. We shall perform them in the case when $\mathbb{Q}_{11}\cup\mathbb{Q}_{13}\neq\varnothing$. Then we also have $\mathbb{Q}_{12}\ne \varnothing$ since otherwise $\mathfrak{P}_1$ and $\mathfrak{P}_2$ are certainly separated.
Every element $\mathcal{Q}_1^i\in\mathbb{Q}_{11}\cup\mathbb{Q}_{12}\cup\mathbb{Q}_{13}$ is of the form
$$
\begin{equation}
\mathcal{Q}_1^i:=\begin{cases} \mathcal{P}_1^i+\mathcal{T}(\mathcal{P}_1^i) &\text{if } \mathcal{T}(\mathcal{P}_1^i)\neq\mathcal{P}_1^i, \\ \mathcal{P}_1^i &\text{if } \mathcal{T}(\mathcal{P}_1^i)= \mathcal{P}_1^i. \end{cases}
\end{equation}
\tag{4.28}
$$
For every vector $\beta_1^i$ in the set $B_1$, we have a projector $\mathcal{P}_1^i=P_1^i \oplus O_2\oplus{O}_3=\langle\,{\cdot}\,, \beta^i_1\rangle\beta^i_1\oplus O_2 \oplus O_3$ which in its turn determines the corresponding projector $\mathcal{Q}_1^i$ of the form (4.28). This enables us to define a map $\mathbf{b}\colon B_1\to\mathbb{Q}_{11}\cup\mathbb{Q}_{12}\cup\mathbb{Q}_{13}$ by the rule Note that if $\mathcal{Q}_1^i=\mathcal P^i_1+\mathcal T(\mathcal P^i_1)=\mathcal P^i_1+\mathcal P^{i'}_1\in\mathbb{Q}_{11}$, then there are two vectors $\beta_1^i$, $\beta_1^{i'}$ such that $\mathbf{b}(\beta_1^i)=\mathbf{b}(\beta_1^{i'})=\mathcal{Q}_1^i=\mathcal{Q}_1^{i'}$. In what follows, $\mathbf{b}^{-1}(\,\cdot\,)$ stands for the full preimage.
By the hypothesis of the lemma, $B_1$ is an equivalence class with respect to $\sim$. Since $\mathbb{Q}_{11}\cup\mathbb{Q}_{13}\neq \varnothing$, there is a pair of vectors $\beta_1^i\in\mathbf{b}^{-1}(\mathbb{Q}_{12})$ and $\beta_1^{i'}\in\mathbf{b}^{-1}(\mathbb{Q}_{11}\cup\mathbb{Q}_{13})$ such that $\langle\beta_1^i,\beta_1^{i'}\rangle\neq 0$. Indeed, the absence of such a pair would mean that
$$
\begin{equation*}
\operatorname{span}B_1=\operatorname{span}\mathbf{b}^{-1}(\mathbb Q_{12})\oplus\,\operatorname{span}\mathbf{b}^{-1}(\mathbb Q_{11}\cup\mathbb Q_{13}),
\end{equation*}
\notag
$$
which is impossible by the definition of the equivalence ${\sim}$.
This pair of vectors determines the projectors $\mathcal{Q}_1^i= \mathcal P^i_1+\mathcal T(\mathcal P^i_1)=\mathbf{b}(\beta^i_1)\in\mathbb Q_{12}$ and $\mathcal{Q}_1^{i'}= \mathbf{b}(\beta^{i'}_1)\in\mathbb{Q}_{11}\cup\mathbb{Q}_{13}$ as well as an element
$$
\begin{equation*}
\widetilde{\mathcal{Q}}_1^i:= \mathcal{Q}_1^i\mathcal{Q}_1^{i'}\mathcal{Q}_1^i\in \mathfrak{Q}.
\end{equation*}
\notag
$$
By (4.22) we have the representations
$$
\begin{equation*}
\begin{gathered} \, \mathcal{Q}_1^i = (\mathcal{Q}_1^i)_1\oplus(\mathcal{Q}_1^i)_2 \oplus (\mathcal{Q}_1^i)_3,\qquad \mathcal{Q}_1^{i'} = (\mathcal{Q}_1^{i'})_1\oplus(\mathcal{Q}_1^{i'})_2 \oplus (\mathcal{Q}_1^{i'})_3, \\ \widetilde{\mathcal{Q}}_1^i = (\widetilde{\mathcal{Q}}_1^i)_1 \oplus(\widetilde{\mathcal{Q}}_1^i)_2 \oplus (\widetilde{\mathcal{Q}}_1^i)_3, \end{gathered}
\end{equation*}
\notag
$$
where $(\mathcal{Q}_1^i)_k,(\mathcal{Q}_1^{i'})_k,(\widetilde{\mathcal{Q}}_1^i)_k\in\mathfrak{P}_k$ for $k=1,2,3$ and, moreover,
$$
\begin{equation*}
(\widetilde{\mathcal{Q}}_1^i)_k = (\mathcal{Q}_1^i)_k(\mathcal{Q}_1^{i'})_k(\mathcal{Q}_1^i)_k, \qquad k=1,2,3.
\end{equation*}
\notag
$$
By the choice of the vector $\beta^i_1$ we have $\mathcal{Q}_1^i=\mathbf{b}(\beta^i_1)\in\mathbb{Q}_{12}$. Therefore, $(\mathcal{Q}_1^i)_3=O_3$ and hence $(\widetilde{\mathcal{Q}}_1^i)_3=O_3$. Similarly, if $\mathcal{Q}_1^{i'}\in\mathbb{Q}_{11}\cup\mathbb{Q}_{13}$, then $(\mathcal{Q}_1^{i'})_2=O_2$, hence $(\widetilde{\mathcal{Q}}_1^i)_2=O_2$. Thus, only the component $(\widetilde{\mathcal{Q}}_1^i)_1$ can be non-zero. In this situation, there are two possibilities.
1) Suppose that $\mathcal{Q}_1^{i'}\in\mathbb{Q}_{13}$. Then $(\mathcal{Q}_1^{i'})_1= P_1^{i'}$ and $(\mathcal{Q}_1^i)_1= P_1^i$, and $(\widetilde{\mathcal{Q}}_1^i)_1$ is of the form
$$
\begin{equation}
(\widetilde{\mathcal{Q}}_1^i)_1 = P_1^i P_1^{i'}P_1^i = \langle\beta_1^i,\beta_1^{i'}\rangle^2 P_1^i.
\end{equation}
\tag{4.29}
$$
2) Suppose that $\mathcal{Q}_1^{i'}\in\mathbb{Q}_{11}$. If $\mathcal{Q}_1^{i'}=\mathcal{P}_1^{i'} $, then we still arrive at (4.29) by arguing as above. Consider the case when $\mathcal{Q}_1^{i'}=\mathcal{P}_1^{i'} + \mathcal{T}(\mathcal{P}_1^{i'})$, where $\mathcal{T}(\mathcal{P}_1^{i'})= \mathcal{P}_1^j\in\mathbb{P}_1$ and $\mathcal{P}_1^j\mathcal{P}_1^{i'}=O_1$. The projector $\mathcal{P}_1^j$ corresponds to the vector $\beta_1^j\in B_1$. Then $(\widetilde{\mathcal{Q}}_1^i)_1$ takes the form
$$
\begin{equation}
(\widetilde{\mathcal{Q}}_1^i)_1 = P_1^i (P_1^{i'}+P_1^j)P_1^i = [\langle\beta_1^i,\beta_1^{i'}\rangle^2+\langle\beta_1^i,\beta_1^j\rangle^2]P_1^i.
\end{equation}
\tag{4.30}
$$
Comparing (4.29) with (4.30), we arrive at the following relation:
$$
\begin{equation*}
\widetilde{\mathcal{Q}}_1^i = c \mathcal{P}_1^i,\qquad c\geqslant\langle\beta_1^i,\beta_1^{i'}\rangle^2>0.
\end{equation*}
\notag
$$
This means that the algebra $\mathfrak{Q}$ contains separately the projectors $\mathcal{P}_1^i\in\mathbb{P}_1$ and $\mathcal{T}(\mathcal{P}_1^i)\in\mathbb{P}_2$. We note in this connection that the projector $\mathcal Q_1^i=\mathcal{P}_1^i+\mathcal{T}(\mathcal{P}_1^i)$ splits into independent (in $\mathfrak{Q}$) one-dimensional parts $\mathcal{P}_1^i$ and $\mathcal{T}(\mathcal{P}_1^i)$.
We now consider the map $\mathcal T'\colon {\mathcal{P}}\to\mathcal{T}$, which differs from $\mathcal{T}$ by its values on only two projectors: $\mathcal{P}_1^i$ and $\mathcal{T}(\mathcal{P}_1^i)$. Put
$$
\begin{equation*}
\mathcal{T}'(\mathcal{P}_1^i):=\mathcal{P}_1^i,\qquad \mathcal{T}'(\mathcal{P}_1^i):=\mathcal{T}(\mathcal{P}_1^i).
\end{equation*}
\notag
$$
Then the algebras $\mathfrak{Q}$ and $\mathfrak{Q}'$ defined by the maps $\mathcal{T}$ and $\mathcal{T}'$ obviously coincide, and $\mathfrak{Q}'$ has its own representation of the form (4.26):
$$
\begin{equation*}
\mathfrak{Q}'=\vee[\mathbb{Q}_{11}'\cup\mathbb{Q}_{22}'\cup\mathbb{Q}_{33}'\cup\mathbb{Q}_{12}' \cup\mathbb{Q}_{13}'\cup\mathbb{Q}_{23}']=\mathfrak{Q},
\end{equation*}
\notag
$$
whose relation to (4.26) is as follows:
$$
\begin{equation*}
\begin{gathered} \, \mathbb{Q}_{11}'=\mathbb{Q}_{11}\cup\{\mathcal{P}_1^i\}, \qquad \mathbb{Q}_{22}'=\mathbb{Q}_{22}\cup\{\mathcal{T}(\mathcal{P}_1^i)\}, \qquad \mathbb{Q}_{12}'=\mathbb{Q}_{12}\setminus\{\mathcal{Q}_1^i\}, \\ \mathbb{Q}_{13}'=\mathbb{Q}_{13},\qquad\mathbb{Q}_{23}'=\mathbb{Q}_{23}, \qquad\mathbb{Q}_{33}'=\mathbb{Q}_{33}. \end{gathered}
\end{equation*}
\notag
$$
Thus, the splitting of $\mathcal Q^i_1$ results in a decrease of the part $\mathbb Q_{12}$, which is responsible for linking the blocks $\mathfrak{P}_1$ and $\mathfrak{P}_2$, by one projector.
The same argument for the part $\mathbb{Q}_{12}'\subset\mathbb{Q}_{12}$ yields that we can remove another projector from it without changing the algebra $\mathfrak{Q}$. Repeating this procedure finitely many times, we remove all the projectors contained in $\mathbb{Q}_{12}$ and hence obtain the representation (4.27). Thus we have shown that the condition $\mathbb{Q}_{11}\cup\mathbb{Q}_{13}\cup\mathbb{Q}_{22}\cup\mathbb{Q}_{23}= \varnothing$ is necessary for the algebra to link the blocks $\mathfrak{P}_1$ and $\mathfrak{P}_2$. Note that the condition $\mathbb{Q}_{11}\cup\mathbb{Q}_{13}=\varnothing$ (resp. $\mathbb{Q}_{22}\cup\mathbb{Q}_{23}=\varnothing$) is equivalent to the inclusion $\mathcal{T}(\mathbb{P}_1)\subset\mathbb{P}_2$ (resp. $\mathcal{T}(\mathbb{P}_2)\subset\mathbb{P}_1$). Since $\mathcal T$ is an involution, it follows that $\mathbb{Q}_{11}\cup\mathbb{Q}_{13}\cup\mathbb{Q}_{22}\cup\mathbb{Q}_{23}= \varnothing$ if and only if $\mathcal{T}(\mathbb{P}_1)=\mathbb{P}_2$. 4. From now on, we assume that $\mathcal{T}(\mathbb{P}_1)=\mathbb{P}_2$ and, therefore, the algebra $\mathfrak{Q}$ connecting $\mathfrak{P}_1$ and $\mathfrak{P}_2$ can be written in the form
$$
\begin{equation*}
\mathfrak{Q}=\vee[\mathbb{Q}_{12}\cup\mathbb{Q}_{33}].
\end{equation*}
\notag
$$
Hence we see that $\mathfrak{Q}=\mathfrak{Q}_{12}\oplus\mathfrak{Q}_{3} $, where $\mathfrak{Q}_{12}:=\vee \mathbb{Q}_{12}$, $\mathfrak{Q}_{3} =\vee \mathbb{Q}_{33}$ and the algebra $\mathfrak{Q}$ separates the blocks $\mathfrak{Q}_{12}$ and $\mathfrak{Q}_3$.
We now specify the structure of the algebra $\mathfrak{Q}_{12}$. For this purpose, it is convenient to use the matrix notation
$$
\begin{equation*}
\mathfrak{Q}_{12} = \vee\biggl\{\mathcal{Q}^i:= \begin{pmatrix} P_1^i &\\ & P_2^{i'} \end{pmatrix}\biggm| \mathcal{T}(\mathcal{P}_1^i)=\mathcal{P}_2^{i'};\, i=1,\dots,n_1 \biggr\}
\end{equation*}
\notag
$$
(with zero entries omitted) and the representation
$$
\begin{equation*}
\mathcal{Q}^i= \begin{pmatrix} \langle\,{\cdot}\,,\beta_1^i\rangle\beta_1^i & \\ & \langle\,{\cdot}\,,\beta_2^{i'}\rangle\beta_2^{i'} \end{pmatrix}
\end{equation*}
\notag
$$
in terms of the vectors $\beta_k^i$ corresponding to the projectors $P_1^i$ and $P_2^{i'}$. Now suppose that ${G}(\mathbb{P}_1)\neq{G}(\mathcal{T}(\mathbb{P}_1))$. Note that this is possible only when both $B_1$ and $B_2$ consist of more than one element. Under this assumption, the algebra $\mathfrak{Q}_{12}$ contains $\mathcal{Q}^i$ and $\mathcal{Q}^j$ such that $|\langle\beta_1^i,\beta_1^j\rangle\neq\langle\beta_2^{i'},\beta_2^{j'}\rangle|$. For the product $\mathcal{Q}^i\mathcal{Q}^j\mathcal{Q}^i\in\mathfrak{Q}_{12}$ we have a representation
$$
\begin{equation*}
\mathcal{Q}^i\mathcal{Q}^j\mathcal{Q}^i= \begin{pmatrix} \langle\beta_1^i,\beta_1^j\rangle^2 \langle\,{\cdot}\,,\beta_1^i\rangle \beta_1^i&\\ & \langle\beta_2^{i'},\beta_2^{j'}\rangle^2\langle\,{\cdot}\,, \beta_2^{i'}\rangle\beta_2^{i'} \end{pmatrix}
\end{equation*}
\notag
$$
and arrive at the relations
$$
\begin{equation*}
\begin{gathered} \, \mathcal{Q}^i = \begin{pmatrix} P_1^i &\\ & O_2\end{pmatrix}+ \begin{pmatrix} O_1 &\\ & P_2^{i'}\end{pmatrix}, \\ \mathcal{Q}^i\mathcal{Q}^j\mathcal{Q}^i = \langle\beta_1^i,\beta_1^j\rangle^2\begin{pmatrix} P_1^i &\\ & O_2 \end{pmatrix}+ \langle\beta_2^{i'},\beta_2^{j'}\rangle^2\begin{pmatrix} O_1 &\\ & P_2^{i'}\end{pmatrix}. \end{gathered}
\end{equation*}
\notag
$$
Combining these with the inequality $|\langle\beta_1^i,\beta_1^j\rangle|\neq|\langle\beta_2^{i'},\beta_2^{j'}\rangle|$, we conclude that the algebra $\mathfrak{Q}_{12}$ contains the projectors
$$
\begin{equation*}
\begin{pmatrix}P_1^i &\\& O_2 \end{pmatrix} \quad\text{and}\quad \begin{pmatrix} O_1 &\\ & P_2^{i'}\end{pmatrix}.
\end{equation*}
\notag
$$
Hence the algebra $\mathfrak{Q}$ separates $\mathfrak{P}_1$ and $\mathfrak{P}_2$, and the following representation holds:
$$
\begin{equation*}
\mathfrak{Q}=\mathfrak{P}_1\oplus\mathfrak{P}_2\oplus \mathfrak{Q}_3.
\end{equation*}
\notag
$$
5. Thus the condition ${G}(\mathbb{P}_1)={G}(\mathcal{T}(\mathbb{P}_1))$ is also necessary for the algebra $\mathfrak{Q}$ to connect the blocks $\mathfrak{P}_1$ and $\mathfrak{P}_2$. Under this condition, we have the representation
$$
\begin{equation*}
\mathfrak{Q}_{12}=\vee\biggl\{ \begin{pmatrix} \mathcal P &\\\ & \mathcal{T}(\mathcal P) \end{pmatrix} \biggm| \mathcal P\in\mathbb{P}_1 \biggr\}=\vee\{\mathcal P\oplus\mathcal T(\mathcal P)\mid \mathcal P\in\mathbb P_1\}.
\end{equation*}
\notag
$$
This completes the proof of Theorem 1. $\Box$ We point out an important circumstance. Under the hypotheses of Theorem 1, the existence of a connection between $\mathfrak{P}_1$ and $\mathfrak{P}_2$ excludes connection between any of these blocks and $\mathfrak{P}_3$. This follows from the italicized remark on the separation of $\mathfrak{Q}_{12}$ and $\mathfrak{Q}_3$ at the beginning of part 4 of the proof. The following result of Korikov [26] enables us to clarify the structure of the algebra $\mathfrak{Q}_{12}$ in Theorem 1. Suppose that $\mathcal{T}(\mathbb{P}_1)=\mathbb{P}_2$ and accordingly $n_1=n_2=:n$. We enumerate these sets consistently:
$$
\begin{equation*}
\mathbb{P}_1=\{\mathcal{P}_1^i\mid i=1,\dots,{n}\},\qquad \mathbb{P}_2=\{\mathcal{P}_2^i\mid \mathcal{P}_2^i=\mathcal{T}(\mathcal{P}_1^i);\,i=1,\dots,n\}.
\end{equation*}
\notag
$$
With every projector $\mathcal{P}_k^i$ we associate a one-dimensional subspace $L_k^i$:
$$
\begin{equation*}
L_k^i:= P_k^i \mathscr{G}_k = \operatorname{span}\{\beta_k^i\}\subset \mathscr{G}_k.
\end{equation*}
\notag
$$
The angle between subspaces $L$ and $M$ is defined by the relation
$$
\begin{equation*}
\phi(L,M):=\arccos \|P_{L}P_{M}\|\in\biggl[0,\frac{\pi}2\biggr],
\end{equation*}
\notag
$$
where $P_{L}$, $P_{M}$ are the corresponding orthogonal projectors. With families of subspaces we associate the following families of angles:
$$
\begin{equation*}
\begin{alignedat}{2} \varphi^i_k &:=\phi(L^i_k,L^1_k+\dots+L^{i-1}_k), &\qquad i &=1,\dots,n, \\ \varphi^{ij}_k &:=\phi(L^i_k,L^j_k), &\qquad i,j &=1,\dots,n,\, i<j, \\ \varphi^{ij,l}_k &:=\phi(L^i_k+L^j_k,L^l_k), &\qquad i,j,l &=1,\dots,n,\, i<j<l. \end{alignedat}
\end{equation*}
\notag
$$
A straightforward application of the unitary equivalence criterion for families of subspaces in [26] leads to the following result. Lemma 4. Suppose that the hypotheses of Theorem 1 hold for $\mathbb{P}_1$ and $\mathbb{P}_2$. Then the map $\mathcal{T}$ extends from the generators $\mathcal{P}_k^i$ to an algebra isomorphism $\mathcal{I}\colon\mathfrak{P}_1\to\mathfrak{P}_2$, and the algebra $\mathfrak{Q}_{12}$ takes the form if and only if the following equalities hold for all $i$, $j$, $l$: It is also easy to show that if at least one equality in Lemma 4 does not hold, then the algebra $\mathfrak{Q}_{12}$ does not connect $\mathfrak{P}_1$ and $\mathfrak{P}_2$. Thus, simultaneous fulfillment of the conditions in Theorem 1 and Lemma 4 guarantees that the blocks are connected and there is an isomorphism $\mathcal{I}$, while lack of at least one of these conditions leads to separation of the corresponding blocks by the algebra $\mathfrak{Q}$. Consider the more general situation when there are $N$ Hilbert spaces $\mathscr{G}_k$, $k=1,\dots,N$. Each $\mathscr{G}_k$ has its own set of one-dimensional projectors
$$
\begin{equation*}
\mathbb P_k:=\{P^1_k,\dots,P^{n_k}_k\},\qquad P^i_k=\langle\,{\cdot}\,,\beta^i_k\rangle\,\beta^i_k,\quad \|\beta^i_k\|=1,
\end{equation*}
\notag
$$
defined by the set of vectors $B_k:=\{\beta^i_k\mid i=1,\dots,{n}_k\}\subset \mathscr{G}_k$, and each $B_k$ is an equivalence class with respect to $\sim$. These projectors generate the algebras
$$
\begin{equation*}
\mathfrak{P}_k:=\vee\mathbb P_k=\vee\{P^1_k,\dots,P_k^{n_k}\} \cong \mathbb{M}^{l_k},
\end{equation*}
\notag
$$
where $l_k:=\operatorname{dim}\operatorname{span}B_k$. We consider the algebra with generators
$$
\begin{equation*}
\mathcal{P}_k^i:= O_1\oplus \dots \oplus {O}_{k-1} \oplus P_k^i \oplus{O}_{k+1}\oplus\dots \oplus O_N,
\end{equation*}
\notag
$$
where ${O}_k$ is the zero operator acting on the $k$th component of the space $\bigoplus\sum_{k=1}^N\mathscr{G}_k$. The components $\mathfrak{P}_k$ in (4.31) are referred to as blocks of the algebra $\mathfrak{P}_*$. We endow the total set of generators
$$
\begin{equation*}
\mathbb{P}:=\mathbb P_1\cup\dots\cup\mathbb P_N=\{\mathcal{P}_k^i\mid i=1,\dots,{n}_k;\,k=1,\dots, N\}
\end{equation*}
\notag
$$
of the algebra $\mathfrak P$ with a map $\mathcal{T}\colon\mathbb{P}\to\mathbb{P}$ such that if $\mathcal{T}(\mathcal{P}_k^i)=\mathcal{P}_{k'}^{i'}$, then $\mathcal{T}(\mathcal{P}_{k'}^{i'})=\mathcal{P}_{k}^i$ and (exactly) one of the following conditions holds:
$$
\begin{equation*}
\mathcal{T}(\mathcal{P}_k^i)=\mathcal{P}_k^i\quad\text{or}\quad \mathcal{T}(\mathcal{P}_k^i)\mathcal{P}_k^i=\mathcal{P}_k^i\mathcal{T}(\mathcal{P}_k^i)=0.
\end{equation*}
\notag
$$
Using this map, we define the projectors
$$
\begin{equation*}
\mathcal{Q}_k^i := \begin{cases} \mathcal{P}_k^i &\text{if }\mathcal{T}(\mathcal{P}_k^i)=\mathcal{P}_k^i, \\ \mathcal{P}_k^i+\mathcal{T}(\mathcal{P}_k^i) &\text{if } \mathcal{T}(\mathcal{P}_k^i)\mathcal{P}_k^i=0, \end{cases}
\end{equation*}
\notag
$$
some of which may coincide. We construct the algebra
$$
\begin{equation}
\mathfrak{Q}:=\vee\{\mathcal{Q}_k^i\mid i=1,\dots,{n}_k;\,k=1,\dots, N\}\subset \mathfrak{P}
\end{equation}
\tag{4.32}
$$
and describe its structure using the results of Theorem 1 and Lemma 4. We form all possible pairs $\{\mathfrak{P}_k, \mathfrak{P}_{k'}\}$, $k\ne k'$, of the blocks constituting the algebra $\mathfrak{P}$ in (4.31) and choose the pairs connected by the algebra $\mathfrak{Q}$ (in the same sense as $\mathfrak{P}_1$, $\mathfrak{P}_2$ in Theorem 1 and Lemma 4). This choice is unique because, as noted after the proof of the theorem, each $\mathfrak{P}_k$ can be connected with only one $\mathfrak{P}_{k'}$. We renumber the blocks in (4.31), distinguishing the pairs of connected blocks and independent blocks:
$$
\begin{equation}
\underbrace{\mathfrak{P}_1, \mathfrak{P}_{2}};\,\dots;\,\underbrace{\mathfrak{P}_{2k-1}, \mathfrak{P}_{2k}};\,\dots;\,\underbrace{\mathfrak{P}_{2N_1-1}, \mathfrak{P}_{2N_1}};\,\mathfrak{P}_{2N_1+1};\,\dots;\, \mathfrak{P}_{2N_1+j};\,\dots;\,\mathfrak{P}_{N},
\end{equation}
\tag{4.33}
$$
and grouping the components of the set $\mathbb P$ accordingly:
$$
\begin{equation*}
\underbrace{\mathbb{P}_1, \mathbb{P}_{2}};\,\dots;\,\underbrace{\mathbb{P}_{2k-1}, \mathbb{P}_{2k}};\,\dots; \,\underbrace{\mathbb{P}_{2N_1-1}, \mathbb{P}_{2N_1}};\,\mathbb{P}_{2N_1+1};\, \dots;\, \mathbb{P}_{2N_1+j};\,\dots;\, \mathbb{P}_{N}.
\end{equation*}
\notag
$$
This numbering will be used hereafter. We see that this grouping reduces the map $\mathcal T$ in the following sense:
$$
\begin{equation}
\begin{gathered} \, \mathcal T(\{\mathbb P_{2k-1},\mathbb P_{2k}\})=\{\mathbb P_{2k-1},\mathbb P_{2k}\},\quad \mathcal T(\mathbb P_{2k-1})=\mathbb P_{2k},\qquad k=1,\dots,N_1, \\ \mathcal{T}(\mathbb P')=\mathbb P',\quad \text{where}\quad\mathbb P':=\mathbb P_{2N_1+1}\cup\dots\cup\mathbb P_{N}. \end{gathered}
\end{equation}
\tag{4.34}
$$
The blocks $\mathfrak{P}_{2N_1+j}$ are distinguished by the fact that they are pairwise separated (not connected) by the algebra $\mathfrak{Q}$. Ib view of separation, if the projector
$$
\begin{equation*}
\mathcal{Q}_{2N_1+j}^i=\mathcal{P}_{2N_1+j}^i+\mathcal{T}(\mathcal{P}_{2N_1+j}^i)
\end{equation*}
\notag
$$
is such that $\mathcal{T}(\mathcal{P}_{2N_1+j}^i)\notin\mathbb{P}_{2N_1+j}$, then $\mathfrak{Q}$ contains both $\mathcal{P}_{2N_1+j}^i$ and $\mathcal{T}(\mathcal{P}_{2N_1+j}^i)$. This enables us to replace $\mathcal T|_{\mathbb P'}$ with a new map $\mathcal T'\colon {\mathbb P'}\to{\mathbb P'}$ that is defined by $\mathcal{P}_k^i\in\mathbb{P}'$ using the rules
$$
\begin{equation*}
\mathcal T'(\mathcal{P}_k^i):= \begin{cases} \mathcal{P}_k^i &\text{if }\mathcal{T}(\mathcal{P}_k^i)\notin \mathbb{P}_k, \\ \mathcal{P}_k^i &\text{if }\mathcal{T}(\mathcal{P}_k^i)= \mathcal{P}_k^i, \\ \mathcal{T}(\mathcal{P}_k^i) &\text{if }\mathcal{T}(\mathcal{P}_k^i) \in \mathbb{P}_k \text{ and } \mathcal{T}(\mathcal{P}_k^i) \neq \mathcal{P}_k^i. \end{cases}
\end{equation*}
\notag
$$
It is easy to see that the map
$$
\begin{equation*}
\widetilde{\mathcal T}\colon \mathbb P\to\mathbb P,\qquad\widetilde{\mathcal T}:=\begin{cases} \mathcal T &\text{on }\mathbb P\setminus\mathbb P', \\ \mathcal T' &\text{on }\mathbb P', \end{cases}
\end{equation*}
\notag
$$
determines the same algebra $\mathfrak{Q}$ as $\mathcal T$ and, in addition to (4.34), it is reduced by independent blocks $\widetilde{\mathcal T}(\mathbb P_{2N+j})=\mathbb P_{2N+j}$. On these blocks, it either acts identically or sends each projector to its orthogonal. The map $\mathcal T$ determines the algebra ${\mathfrak{Q}}$ in terms of the generating projectors $Q^i_k$ in (4.32). The map $\widetilde{\mathcal T}$ similarly determines the projectors $\widetilde Q^i_k$ generating the same algebra. A special feature of the latter projectors is in their form. By construction, we have
$$
\begin{equation*}
\widetilde{\mathcal{Q}}^i_k=\begin{cases} {\mathcal P}_k^i+\widetilde{\mathcal T}({\mathcal P}_k^i) &\text{if } \widetilde{\mathcal T}({\mathcal P}_k^i)\in\mathbb{P}_k,\ \ \widetilde{\mathcal T}({\mathcal P}_k^i)\neq{\mathcal P}_k^i, \\ {\mathcal P}_k^i &\text{in other cases}, \end{cases}
\end{equation*}
\notag
$$
i.e., all two-dimensional $\widetilde{\mathcal{Q}}^i_k$ are sums of projectors belonging to the same block $\mathfrak{P}_k$. We put $\mathbb{Q}_k:=\{\widetilde{\mathcal{Q}}_k^i\mid i=1,\dots,n_k\}$. The following result summarizes our considerations. Recall that the blocks are numbered according to (4.33). Proposition 8. The algebra $\mathfrak{Q}$ admits a representation where $\mathfrak{Q}_k^{\mathrm{I}}:=\{A\oplus \mathcal TA\mid A\in\mathfrak{P}_{2k-1}\}\subset\mathfrak{P}_{2k-1}\oplus\mathfrak{P}_{2k}$ and $\mathfrak Q^{\mathrm{II}}_k =\vee \mathbb Q_k\subset \mathfrak P_k$. 4.5. The canonical formWe now use the results established above to describe the structure of the eikonal algebra. The description will be given by a canonical form (representation). The grouping of projectors in (4.16) in accordance with the decomposition (4.15) is the partition of the sets $\mathbb P_{\Phi^j}$ into the equivalence classes $\mathbb P_{\Phi^j}^k$ with respect to $\sim$ (they are also the classes $\mathbb P_{l}$). It is motivated by Proposition 7 and prepares our application of Theorem 1. The role of the algebra $\mathfrak P$ in Theorem 1 (see (4.31)) is played by the algebra
$$
\begin{equation*}
\mathfrak P^\partial:=\biggl[ \bigoplus_{l=1}^{L} \mathfrak{P}_l\biggr]\oplus\biggl[ \bigoplus_{l=1}^{L}\mathfrak{P}_l\biggr].
\end{equation*}
\notag
$$
It consists of $2L$ irreducible blocks and is represented as $\mathfrak P^\partial=\vee \mathbb P^\partial$, where
$$
\begin{equation*}
\begin{aligned} \, {\mathbb P}^\partial &:= \{\mathcal{P}_{\gamma l}^k\oplus O\mid k=1,\dots,n_{\gamma l};\, l=1,\dots,L;\, \gamma\in\Sigma\} \\ &\qquad\cup\{ O\oplus\mathcal{P}_{\gamma l}^k\mid k=1,\dots,n_{\gamma l};\, l=1,\dots,L;\, \gamma\in\Sigma\} \end{aligned}
\end{equation*}
\notag
$$
where $\mathcal{P}_{\gamma l}^k$ are the projectors in (4.19), $O$ is the zero element of the algebra $\bigoplus_{l=1}^{L} \mathfrak{P}_l$. Putting
$$
\begin{equation*}
\mathcal P_{\gamma l}^{k r}:= \begin{cases} \mathcal{P}_{\gamma l}^k\oplus O, &r= 0, \\ O\oplus\mathcal{P}_{\gamma l}^k, &r =\varepsilon_l, \end{cases}
\end{equation*}
\notag
$$
we have
$$
\begin{equation*}
{\mathbb P}^\partial= \{\mathcal{P}_{\gamma l}^{k r}\mid k=1,\dots,n_{\gamma l},\, l=1,\dots,L,\, r=0,\varepsilon_l; \, \gamma\in\Sigma\}.
\end{equation*}
\notag
$$
We define a map $\mathcal{T}\colon {\mathbb P}^\partial\to{\mathbb P}^\partial$ in terms of the formalism specifying the relations between the triples $(k,l,r_l)$:
$$
\begin{equation}
\mathcal{T}(\mathcal P_{\gamma l}^{k r_{l}}):= \begin{cases} \mathcal{P}_{\gamma l'}^{k' r_{l'}} &\text{if }(k',l',r_{l'})\leftrightarrow(k,l,r_l), \\ \mathcal{P}_{\gamma l}^{k r_l} &\text{if } (k,l,r_l)\text{ not related to any } (k',l',r_{l'}). \end{cases}
\end{equation}
\tag{4.36}
$$
It is not difficult to check that the set $\mathbb P^\partial$ and the map $\mathcal T$ satisfy all the hypotheses of Theorem 1, and that the algebra $\mathfrak Q$ determined by them in accordance with (4.25) coincides with the boundary algebra (4.20):
$$
\begin{equation*}
\mathfrak Q=\partial ({U}{\mathfrak{E}}^T_\Sigma U^{-1}).
\end{equation*}
\notag
$$
To write the eikonal algebra in the canonical form, we use junction of its blocks connected by the boundary algebra. Let us describe this construction. Let $\mathfrak{A}=\dot C([0,\varepsilon];\mathfrak{P})$ and $\mathfrak{B}=\dot C([0,\varepsilon'];\mathfrak{P}')$ be standard algebras such that
$$
\begin{equation*}
\mathfrak{A}(\varepsilon):=\{a(\varepsilon)\mid a\in\mathfrak{A}\}=\mathfrak{P},\qquad \mathfrak{B}(0):=\{b(0)\mid b\in\mathfrak{B}\}=\mathfrak{P}',
\end{equation*}
\notag
$$
and $\mathfrak{P}\cong\mathfrak{P}'$ via an isomorphism $\mathcal I\colon \mathfrak{P}'\to\mathfrak{P}$. We define an algebra
$$
\begin{equation*}
\mathfrak{A}^{\oplus}\mathfrak{B}:= \{a\oplus b\mid a\in\mathfrak{A},\, b\in\mathfrak{B}, \, a(\varepsilon)=\mathcal I b(0)\}.
\end{equation*}
\notag
$$
Algebras of this kind arise in the course of transforming to the canonical form in the situation when the representation (4.35) of the boundary algebra $\partial (U\mathfrak{E}^T_\Sigma U^{-1})\,{=}\,\mathfrak Q$ contains blocks of type $\mathfrak{Q}_{k}^I$ connecting the boundary values of $\mathfrak{A}(\varepsilon)$ and $\mathfrak{B}(0)$ for some pair of blocks $\mathfrak{A}$ and $\mathfrak{B}$ of the eikonal algebra. Given any elements $a\in\mathfrak{A}$ and $b\in\mathfrak{B}$ with $a(\varepsilon)=\mathcal I b(0)$, we define an element $a\sqcup b\in C([0,\varepsilon+\varepsilon'],\mathfrak{P})$ by the rule
$$
\begin{equation}
(a\sqcup b)(r):=\begin{cases} a(r), &r\in[0,\varepsilon], \\ \mathcal I b(r-\varepsilon), &r\in[\varepsilon,\varepsilon+\varepsilon']. \end{cases}
\end{equation}
\tag{4.37}
$$
We call it the junction of $a$ and $b$. Then we define the junction of algebras $\mathfrak{A}$ and $\mathfrak{B}$:
$$
\begin{equation*}
\mathfrak{A}\sqcup\mathfrak{B}:= \{a\sqcup b\in C([0,\varepsilon+\varepsilon'];\mathfrak{P})\mid a\in\mathfrak{A},\, b\in\mathfrak{B},\, a(\varepsilon)=\mathcal I b(0) \}.
\end{equation*}
\notag
$$
We see that $\mathfrak{A}\sqcup\mathfrak{B}$ is a subalgebra of $C([0,\varepsilon+\varepsilon'];\mathfrak{P})$, which is a standard algebra satisfying the representation
$$
\begin{equation*}
\mathfrak{A}\sqcup\mathfrak{B}=\{c\in C([0,\varepsilon+\varepsilon'];\mathfrak{P})\mid c(0)\in\mathfrak{A}(0),\, c(\varepsilon+\varepsilon')\in\mathcal I [\mathfrak{B}(\varepsilon')]\}
\end{equation*}
\notag
$$
and the equalities
$$
\begin{equation*}
(\mathfrak{A}\sqcup\mathfrak{B})(0) = \mathfrak{A}(0),\qquad (\mathfrak{A}\sqcup\mathfrak{B})(\varepsilon+\varepsilon') = \mathfrak{B}(\varepsilon').
\end{equation*}
\notag
$$
Note that the algebras $\mathfrak{A}^{\oplus}\mathfrak{B}$ and $\mathfrak{A}\sqcup\mathfrak{B}$ are isomorphic. To summarize, we say that the algebras $\mathfrak{A}$ and $\mathfrak{B}$ admit a junction through the ends $r=\varepsilon$ and $r'=0$. Note that the algebras $\mathfrak{A}(0)$ and $\mathfrak{B}(\varepsilon')$ do not influence the possibility of junction and its result. Obvious changes in (4.37) enable us to define the junction $\mathfrak{A}\,\sqcup\,\mathfrak{B}\,{\subset}\, C([0,\varepsilon+ \varepsilon'];\mathfrak{P})$
$$
\begin{equation}
\begin{aligned} \, &\text{through the ends }r=\varepsilon \text{ and } r'=\varepsilon' \text{ if } \mathfrak{A}(\varepsilon)=\mathcal I[\mathfrak{B}(\varepsilon')]; \\ &\text{through the ends }r=0 \text{ and } r'=\varepsilon' \text{ if } \mathfrak{A}(0)=\mathcal I[\mathfrak{B}(\varepsilon')]; \\ &\text{through the ends } r=0 \text{ and } r'=0 \text{ if } \mathfrak{A}(0)=\mathcal I[\mathfrak{B}(0)]. \end{aligned}
\end{equation}
\tag{4.38}
$$
We endow the algebra $\mathfrak{A}=\dot C([0,\varepsilon],\mathfrak{P})$ with a transposition $t\colon a\mapsto a^t$, $a^t(r): =a(\varepsilon-r)$, $r\in[0,\varepsilon]$, and put $\mathfrak{A}^t:=\{a^t\mid a\in\mathfrak{A}\}$. An isomorphism ${\mathcal M}\colon \mathfrak{P}\to\mathfrak{P}$ induces a transformation $\check{\mathcal M}\colon C([0, \varepsilon],\mathfrak{P})\to C([0,\varepsilon],\mathfrak{P})$, $(\check{\mathcal M}a)(r): =\mathcal M [a(r)]$, $r\in[0,\varepsilon]$. For a standard algebra $\mathfrak{A}$, we put $\check{\mathcal M}\mathfrak{A}:=\{\check{\mathcal M} a\mid a\in\mathfrak{A}\}\subset C([0,\varepsilon],\mathfrak{P})$. The algebras $\mathfrak{A}^t$ and $\mathcal M\mathfrak{A}$ are also standard. We see that $\check{\mathfrak{A}}\cong{\mathfrak{A}}^t\cong{\mathfrak{A}}$ and
$$
\begin{equation*}
\mathfrak{A}^t(0)=\mathfrak{A}(\varepsilon),\quad \mathfrak{A}^t(\varepsilon)=\mathfrak{A}(0); \qquad (\check{\mathcal M}\mathfrak{A})(0)= \mathcal M[\mathfrak{A}(0)],\quad (\check{\mathcal M}\mathfrak{A})(\varepsilon)=\mathcal M[\mathfrak{A}(\varepsilon)].
\end{equation*}
\notag
$$
We are now ready to describe the transformation of the eikonal algebra to the canonical form. It is given by the following procedure. Step 1. Define an isomorphism $\mathbf{U}_0\colon \mathfrak{E}^T_{\Sigma}\to U\mathfrak{E}^T_\Sigma U^{-1}\stackrel{(4.18)}{\subset} \bigoplus_{l=1}^L C([0,\varepsilon_l],\mathfrak{P}_l)$ by specifying it on the generators:
$$
\begin{equation*}
\mathbf{U}_0 E^T_{\gamma} := U E^T_{\gamma}U^{-1}\stackrel{(4.19)}{=} \bigoplus\sum_{l=1}^L \biggl[\sum_{k=1}^{n_{\gamma l}}\tau^k_{\gamma l} P^k_{\gamma l}\biggr].
\end{equation*}
\notag
$$
The blocks of the algebra $\mathbf{U}_0 \mathfrak{E}^T_{\Sigma}$ are
$$
\begin{equation*}
[\mathbf{U}_0 \mathfrak{E}^T_{\Sigma}]_l:=\vee\biggl\{\sum_{k=1}^{n_{\gamma l}}\tau^k_{\gamma l}\biggm| \gamma\in\Sigma\biggr\}\subset C([0,\varepsilon_l];\mathfrak{P}_l).
\end{equation*}
\notag
$$
Let $[\mathbf{U}_0 \mathfrak{E}^T_{\Sigma}]_l$ and $[\mathbf{U}_0 \mathfrak{E}^T_{\Sigma}]_{l'}$ be two blocks whose boundary values $[\mathbf{U}_0 \mathfrak{E}^T_{\Sigma}]_l(\varepsilon_l)$ and $[\mathbf{U}_0 \mathfrak{E}^T_{\Sigma}]_{l'}(0)$ form one block of type ${\mathfrak{Q}}_k^I$ in the boundary algebra $\partial(\mathbf{U}_0 \mathfrak{E}^T_{\Sigma}):=\partial (U\mathfrak{E}^T_\Sigma U^{-1}) = \mathfrak Q$ (see Proposition 8). In this case, the map $\mathcal T$ (see (4.36)) determines an isomorphism $\mathcal I\colon \mathfrak{P}_{l'}\to\mathfrak{P}_l$:
$$
\begin{equation*}
\mathcal I (P^{k'}_{\gamma l'}) = P^{k}_{\gamma l },\quad \text{if}\quad \mathcal T(\mathcal P^{k' 0}_{\gamma l'})=\mathcal P^{k \varepsilon_l}_{\gamma l }.
\end{equation*}
\notag
$$
Hence the algebra $[\mathbf{U}_0 \mathfrak{E}^T_{\Sigma}]_l\sqcup[\mathbf{U}_0 \mathfrak{E}^T_{\Sigma}]_{l'}$ (the junction of these blocks through the ends $r_l=\varepsilon_l$ and $r_{l'}=0$) is well defined. We easily see that the elements
$$
\begin{equation*}
\bigl( [\mathbf{U}_0 E^T_{\gamma}]_l\sqcup [\mathbf{U}_0 E^T_{\gamma}]_{l'}\bigr)(r) :=\begin{cases} {\displaystyle\sum_{k=1}^{n_{\gamma l}}\tau^k_{\gamma l}(r) P^k_{\gamma l}}, &r\in[0,\varepsilon_l], \\ {\displaystyle\mathcal I \biggl[\sum_{k'=1}^{n_{\gamma l'}}\tau^{k'}_{\gamma l'} (r-\varepsilon_l)P^{k'}_{\gamma l'}\biggr]}, &r\in[\varepsilon_l,\varepsilon_l+\varepsilon_{l'}], \end{cases}
\end{equation*}
\notag
$$
constitute a system of generators of this algebra. By the properties of the map $\mathcal T$, the existence of a connection between the blocks implies that $n_{\gamma l}=n_{\gamma l'}$ for any $\gamma\in\Sigma$, and the junctions $[\mathbf{U}_0 E^T_{\gamma}]_l\sqcup [\mathbf{U}_0 E^T_{\gamma}]_{l'}$ can be represented in the following way:
$$
\begin{equation*}
\bigl( [\mathbf{U}_0 E^T_{\gamma}]_l\sqcup [\mathbf{U}_0 E^T_{\gamma}]_{l'}\bigr)(r) =\sum_{k=1}^{n_{\gamma l}}[\tau^k_{\gamma l }\sqcup\tau^{k'}_{\gamma l}](r)P^k_{\gamma l}, \qquad r\in[0,\varepsilon_l+\varepsilon_{l'}],
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
[\tau^k_{\gamma l}\sqcup\tau^{k'}_{\gamma l' }](r):= \begin{cases} \tau^k_{\gamma l}(r), &r\in[0,\varepsilon_l], \\ \tau^{k'}_{\gamma l'}(r-\varepsilon_l), &r\in[\varepsilon_l,\varepsilon_l+\varepsilon_{l'}]. \end{cases}
\end{equation*}
\notag
$$
The functions $[\tau^k_{\gamma l}\sqcup\tau^{k'}_{\gamma l'}]$ are continuous by the equality $\tau^k_{\gamma l}(\varepsilon_{l})=\tau^{k'}_{\gamma l'}(0)$ determining the relation (which in its turn determines the map $\mathcal T$ and the isomorphism $\mathcal I$) between the triples $(k,l,\varepsilon_{l})$ and $(k',l',0)$. Recall that each $\tau^k_{\gamma l}$ is a linear function of one of the following two kinds:
$$
\begin{equation*}
\text{either }\tau^k_{\gamma l}(r)= t^k_{\gamma l}+r, \ \text{ or } \ \tau^k_{\gamma l}(r) =t^k_{\gamma l }-r,
\end{equation*}
\notag
$$
where $t^k_{\gamma l }=\mathrm{const}\geqslant 0$. To be definite, suppose that $\tau^k_{\gamma l}(r)= t^k_{\gamma l}+r$. Since $\tau^k_{\gamma l}(\varepsilon_{l})=\tau^{k'}_{\gamma l'}(0)$ and since the equality $\tau^k_{\gamma l}(r_{l})=\tau^{k'}_{\gamma l'}(r_{l'})$ is possible only for extreme values of the parameters $r_l$ and $r_{l'}$ (in our case $r_l=\varepsilon_l$, $r_{l'}=0$), it follows that $\tau^{k'}_{\gamma l'}(r)= t^{k'}_{\gamma l'}+r$, with $t^{k'}_{\gamma l'}= t^{k}_{\gamma l}+\varepsilon_l$. This leads to the following equalities:
$$
\begin{equation*}
\begin{aligned} \, [\tau^k_{\gamma l}\sqcup\tau^{k'}_{\gamma l'}](r) &= \begin{cases} t^k_{\gamma l }+r, &r\in[0,\varepsilon_l], \\ (t^{k}_{\gamma l}+\varepsilon_{l})+(r-\varepsilon_l), &r\in[\varepsilon_l,\varepsilon_l+\varepsilon_{l'}], \end{cases} \\ &=\begin{cases} t^k_{\gamma l}+r, &r\in[0,\varepsilon_l], \\ t^{k}_{\gamma l}+r, &r\in[\varepsilon_l,\varepsilon_l+\varepsilon_{l'}], \end{cases} = t^k_{\gamma l }+r,\qquad r\in[0,\varepsilon_l+\varepsilon_{l'}]. \end{aligned}
\end{equation*}
\notag
$$
A similar argument holds in the case when $\tau^k_{\gamma l}(r)=t^k_{\gamma l}-r$. Thus the junction of $\tau^k_{\gamma l}$ and $\tau^{k'}_{\gamma l'}$ is a linear function of the same kind as $\tau^k_{\gamma l}$ and $\tau^{k'}_{\gamma l'}$ themselves. It follows that the junction $[\mathbf{U}_0\mathfrak{E}^T_{\Sigma}]_l\sqcup[\mathbf{U}_0 \mathfrak{E}^T_{\Sigma}]_{l'}$ is a standard algebra with generators of the same kind as in the original blocks. We have studied the case when the boundary values $[\mathbf{U}_0 \mathfrak{E}^T_{\Sigma}]_l(\varepsilon_l)$ and $[\mathbf{U}_0 \mathfrak{E}^T_{\Sigma}]_{l'}(0)$ are related. The cases of other possible connections between the boundary values admitting junction of blocks (see (4.38)), can be studied in a similar way. Step 2. Suppose that the blocks $[\mathbf{U}_0 \mathfrak{E}^T_{\Sigma}]_l$ and $[\mathbf{U}_0 \mathfrak{E}^T_{\Sigma}]_{l'}$ are admissible for junction. We define a map
$$
\begin{equation*}
\mathbf{U}_1\colon\mathfrak{E}^T_{\Sigma}\to\mathbf{U}_1 \mathfrak{E}^T_{\Sigma}\subset\biggl[\bigoplus_{\lambda=1\ (\lambda\neq l,l')}^{L} C([0,\varepsilon_\lambda]; \mathfrak{P}_\lambda)\biggr]\oplus C([0,\varepsilon_{l}+\varepsilon_{l'}];\mathfrak{P}_{l})
\end{equation*}
\notag
$$
by specifying it on the generators:
$$
\begin{equation*}
\mathbf{U}_1 E^T_{\gamma} := \biggl[\bigoplus\sum_{\lambda=1\ (\lambda\neq l,l')}^{L} [\mathbf{U}_0 E^T_{\gamma}]_\lambda\biggr] \oplus \bigl([\mathbf{U}_0 E^T_{\gamma}]_{l}\sqcup [\mathbf{U}_0 E^T_{\gamma}]_{l'}\bigr),\qquad \gamma\in\Sigma.
\end{equation*}
\notag
$$
It is not difficult to see that $\mathbf{U}_1$ is an isomorphism: the algebras $\mathbf{U}_0\mathfrak{E}^T_{\Sigma}$ and $\mathbf{U}_1\mathfrak{E}^T_{\Sigma}$ are isomorphic similarly to the algebras $\mathfrak{A}^{\oplus}\mathfrak{B}$ and $\mathfrak{A}\sqcup\mathfrak{B}$. The isomorphism $\mathbf{U}_1$ sends the eikonal algebra to an algebra of the same kind as $\mathbf{U}_0\mathfrak{E}^T_{\Sigma}$, but with $1$ fewer blocks. The number of blocks of type ${\mathfrak{Q}}_{k}^I$ in the boundary algebra $\partial(\mathbf{U}_1\mathfrak{E}^T_{\Sigma})$ also decreases by $1$. Successively replacing all pairs of related blocks by their junctions in this way, we arrive at an isomorphism $\mathbf{U}_N\colon \mathfrak{E}^T_{\Sigma}\to\mathbf{U}_N\mathfrak{E}^T_{\Sigma}$ such that there are no blocks of type $\mathfrak{Q}_{k}^I$ in the boundary algebra $\partial(\mathbf{U}_N\mathfrak{E}^T_{\Sigma})$. Thus, the map $\mathbf{U}_N$ transforms the eikonal algebra to an algebra of the same structure as $\mathbf{U}_0\mathfrak{E}^T_{\Sigma}$, but with independent blocks. Note also that several (more than two) blocks of the original representation $\mathbf{U}_0\mathfrak{E}^T_{\Sigma}$ can be joined into a single new block in the process of transformation to the representation $\mathbf{U}_N\mathfrak{E}^T_{\Sigma}$. Step 3. At this final step, the irreducibility of the algebras $\mathfrak{P}_l$ is used again. For each $\mathfrak{P}_l$ we find a transformation realizing the isomorphism $\mathfrak{P}_l\,{\cong}\,\mathbb M^{\varkappa_l}$. Then we can define an isomorphism $\mathbf{U}\colon \mathfrak{E}^T_{\Sigma}\,{\to}\, \bigoplus_{l=1}^\mathcal L \dot C([0,\zeta_l];\mathbb{M}^{\varkappa_l})$ by specifying it on generators in the following way:
$$
\begin{equation*}
\mathbf{U} E^T_{\gamma}:=\bigoplus\sum_{l=1}^\mathcal L\biggl[\sum_{k=1}^{s_{\gamma l }}\widetilde\tau_{\gamma l}^k(\cdot_l) \widetilde P_{\gamma l}^k\biggr],\qquad \gamma\in\Sigma,
\end{equation*}
\notag
$$
where $\widetilde P_{\gamma l}^k\in\mathbb{M}^{\varkappa_l}$ are one-dimensional (matrix) projectors, pairwise orthogonal for a fixed vertex $\gamma\in\Sigma$, and $\widetilde\tau_{\gamma l}^k$ are linear functions of one of the following two kinds: either $\widetilde\tau^k_{\gamma l}(r)=\widetilde t^{\,k}_{\gamma l}+r$, or $\widetilde\tau^k_{\gamma l}(r)=\widetilde t^{\,k}_{\gamma l}-r$, $r\in[0,\zeta_l]$, where $\widetilde t^{\,k}_{\gamma l}\geqslant0$ are constants and each $\zeta_l$ is the sum of certain lengths $\varepsilon_k$. Thus, by successive transformations of the original parametric form (4.6), we obtain a form of the same structure, but with independent blocks which are standard algebras. We state the final result. Theorem 2. There is an isomorphism $\mathbf U$ sending the algebra $\mathfrak{E}_{\Sigma}^T$ and its generators (eikonals) to the representation We call a representation (form) of this kind canonical. It is not unique, but we can show that any two such representations differ from each other by renumeration of the blocks $[\mathbf{U}\mathfrak{E}^T_{\Sigma}]_l$, their transposition $[\mathbf{U}\mathfrak{E}^T_{\Sigma}]_l\to[\mathbf{U}\mathfrak{E}^T_{\Sigma}]_l^t$, and any isomorphisms $[\mathbf{U}\mathfrak{E}^T_{\Sigma}]_l\to\check{\mathcal M}[\mathbf{U}\mathfrak{E}^T_{\Sigma}]_l$. The ambiguity associated with transposition obviously corresponds to the two directions in which the argument of $\widetilde\tau_{\gamma l}^k$ can vary on the interval $[0,\delta_l]$ (two possible parameterizations of the $l$th block). Here is a comment on Theorem 2. The original parametric form (4.8) of the eikonal algebra corresponded to a partition of the graph $\Omega^T_\Sigma$ into families $\Phi^1,\dots,\Phi^J$. We conjecture that the transition to the canonical form corresponds to a new partition. This is not proven, but it is supported by well-known examples [7], [8]. § 5. Transformation to the canonical form5.1. Canonical representation and spectraThe main advantage of the canonical form is that complete information about the spectrum of the algebra $\mathfrak{E}^T_\Sigma$ can easily be extracted from it and a number of its invariants are revealed. We proceed to describe them. Let us compare the content of Theorem 2 and Proposition 2. Since the canonical form is non-unique (as mentioned after Theorem 2), we fix the numbering and parameterization of its blocks. We also rewrite this form in a new convenient notation (replacing $\zeta_l$ by $\varepsilon_l$ and $s_{{\gamma l}}$ by $n_{\gamma l}$, and removing $(\,\widetilde{\ }\,)$):
$$
\begin{equation}
\mathbf U \mathfrak{E}_{\Sigma}^T =\bigoplus_{l=1}^\mathcal L \dot C([0,\varepsilon_l]; \mathbb{M}^{\varkappa_l}),\qquad \mathbf{U} E_{\gamma}^T = \bigoplus\sum_{l=1}^\mathcal L\biggl[\sum_{k=1}^{n_{\gamma l}}\tau_{\gamma l}^k P_{\gamma l}^k\biggr], \quad \gamma\in\Sigma.
\end{equation}
\tag{5.1}
$$
Let
$$
\begin{equation*}
\psi_{\gamma l}^k :=\{\tau_{\gamma l}^k(r_l)\mid r_l\in(0,\varepsilon_{l})\}=(\tau^k_{\gamma l}(0),\tau_{\gamma l}^k(\varepsilon_l))
\end{equation*}
\notag
$$
be the time cells corresponding to the canonical representation. The right-hand side of the formula (5.1) for the operator $\mathbf{U} E_{\gamma}^T$ yields its spectrum and, since $\mathbf{U}$ is an isomorphism, the spectrum of the eikonal:
$$
\begin{equation}
\sigma_{\mathrm{ac}}({E}^T_{\gamma}) =[1,T_{1}^{\gamma}]\cup[T_2^{\gamma},T_3^{\gamma}] \cup\dots\cup[T_{N_{\gamma}-1}^{\gamma}, T_{N_{\gamma}}^{\gamma}] =\bigcup_{l=1}^{\mathcal{L}}\bigcup_{k=1}^{n_{\gamma l}}\overline{\psi_{\gamma l}^k},
\end{equation}
\tag{5.2}
$$
where every interval $[T^\gamma_{i-1},T^\gamma_i]$ is in turn covered by cells $\overline{\psi_{\gamma l}^k}$ that either do not overlap or have common ends. By Proposition 2, the same is true for the cells $\overline{\psi^i_{\gamma\Phi}}$ associated with the parametric representation (3.12). Comparing the decompositions (5.2) and (3.13), we conclude that the canonical representation corresponds to a new (canonical) splitting of the eikonal spectrum into time cells. We can show that every new cell consists of the old ones, i.e., the time cells enlarge when we pass to the canonical representation. As mentioned in § 4.1, the spectrum of the standard algebra $\dot C([a,b]; \mathbb M^n)$ consists of (the equivalence classes of) irreducible representations corresponding to the interior points of $[a,b]$ as well as (possibly empty) clusters $\{\widehat\pi^1_a,\dots,\widehat\pi^{n_a}_a\}$ and $\{\widehat\pi^1_b,\dots,\widehat\pi^{n_b}_b\}$ adjacent to the ends of $[a,b]$. Equipped with the Jacobson topology, this spectrum is homeomorphic to a space which may naturally be called a closed interval with split ends. It is described by the following construction (see, e.g., [27]). Consider $n_a$ half-open intervals $[a,b)$ and $n_b$ half-open intervals $(a,b]$ with topology from $\mathbb R$. Identify all their interior points with equal coordinates. The resulting quotient space $\mathscr{S}_{[a,b]}$ consists of the part $S_{(a,b)}$ homeomorphic to $(a,b)$ and two sets of pairwise inseparable points (two clusters) $K_a$ and $K_b$. The clusters consist of $n_a$ and $n_b$ points and correspond to $a$ and $b$ respectively. Each cluster is inseparable from $S_{(a,b)}$. The part $S_{(a,b)}:=\operatorname{int}\mathscr{S}_{[a,b]}$ is the set of interior points, each of which has a neighborhood homeomorphic to an (open) interval of the real axis. By the first representation (5.1), the spectrum of the algebra $\mathbf U \mathfrak{E}_{\Sigma}^T$ is homeomorphic to a disjoint union of closed intervals
$$
\begin{equation*}
\mathscr{S}^T_\Sigma=\mathscr{S}_{[0,\varepsilon_1]}\cup\dots \cup\mathscr{S}_{[0,\varepsilon_\mathcal L]}, \qquad\mathscr{S}_{[0,\varepsilon_l]}: =K^l_{0}\cup S_{(0,\varepsilon_l)}\cup K^l_{\varepsilon_l}.
\end{equation*}
\notag
$$
Every closed interval $\mathscr{S}_{[0,\varepsilon_l]}$ is an arcwise connected component of $\mathscr{S}^T_\Sigma$ and the part $S_{(0,\varepsilon_l)}=\operatorname{int}\mathscr{S}_{[0,\varepsilon_l]}$ is the set of interior points. The spectra of the algebras $\mathfrak{E}_{\Sigma}^T$ and $\mathbf U \mathfrak{E}_{\Sigma}^T$ are connected by the homeomorphism $\mathbf U_*\colon \widehat{\mathfrak{E}_{\Sigma}^T}\to\widehat{\mathfrak{E}_{\Sigma}^T}$ (see (4.2)). Hence, $\widehat{\mathfrak{E}_{\Sigma}^T}$ is homeomorphic to the space $\mathscr{S}^T_\Sigma$ and admits a representation
$$
\begin{equation}
\widehat{\mathfrak{E}_{\Sigma}^T}=\mathscr{S}_1\cup\dots \cup\mathscr{S}_\mathcal L, \qquad \mathscr{S}_l=\eta(\mathscr{S}_{[0,\varepsilon_l]})=\mathscr{K}^l_0\cup S_l\cup\mathscr{K}^l_{\varepsilon_l},
\end{equation}
\tag{5.3}
$$
where $\eta\colon\mathscr{S}^T_\Sigma\to\widehat{\mathfrak{E}_{\Sigma}^T}$ is a homeomorphism, $S_l=\eta(S_{(0, \varepsilon_l)})$, $\mathscr{K}^l_0=\eta(K^l_0)$, $\mathscr{K}^l_{\varepsilon_l}=\eta(K^l_{\varepsilon_l})$. It determines the partition of the spectrum into arcwise connected components and, therefore, has an invariant topological meaning. This is also true for the partition of the closed intervals $\mathscr{S}_l$ into interior points $S_l$ and clusters $\mathscr{K}^l_0$, $\mathscr{K}^l_{\varepsilon_l}$. The set of interior points of the spectrum is $\operatorname{int}\widehat{\mathfrak{E}_{\Sigma}^T}=S_1\cup\dots \cup S_\mathcal L$. 5.2. CoordinatesRecall that the algebras $\mathfrak{E}_{\gamma}^T=\vee\{{{E}^T_{\gamma}}\}\subset\mathfrak{E}_{\Sigma}^T$, which correspond to individual eikonals, are said to be partial. We denote by $\pi$ and $\widehat\pi$ an irreducible representation and its equivalence class. The correspondence $\varphi({E}^T_{\gamma})\leftrightarrow\varphi$ is an algebra isomorphism between $\mathfrak{E}_{\gamma}^T$ and $C(\sigma_{\mathrm{ac}}( E^T_\gamma))$, which is given by the first equality in (4.5) (see [23]). Each $\mathfrak{E}_{\gamma}^T$ is a commutative subalgebra in $\mathfrak{E}_{\Sigma}^T$. Its spectrum (set of characters) $\widehat{\mathfrak{E}_{\gamma}^T}$ is exhausted by Dirac measures:
$$
\begin{equation*}
\widehat{\mathfrak{E}_{\gamma}^T}=\{\widehat{\delta}_t \mid t\in \sigma_{\mathrm{ac}}(E^T_{\gamma})\},\qquad \mathfrak{E}_{\gamma}^T\ni \varphi(E^T_\gamma)\stackrel{\delta_t}{\mapsto} \varphi(t)\in\mathbb R,
\end{equation*}
\notag
$$
where $\widehat\delta_t=\{\delta_t\}$ (see [25], [28]). Thus, for every character we have a point (number) $t$ in the union of intervals (5.2). This point will be regarded as its $\gamma$-coordinate. We also note that any reducible matrix representation of the algebra $\mathfrak{E}_{\gamma}^T$ is of the form
$$
\begin{equation}
\rho \sim\delta_{t_1}\oplus\dots\oplus\delta_{t_p},\qquad \rho(\varphi(E^T_\gamma))\sim\operatorname{diag}\{\varphi(t_1),\dots,\varphi(t_p)\}.
\end{equation}
\tag{5.4}
$$
The numbers $t_1,\dots,t_p$, which uniquely determine the class $\widehat\rho$, are called its $\gamma$-coordinates. Our next step is a coordinatization of the spectrum of the algebra $\mathfrak{E}_{\Sigma}^T$. If a commutative algebra has a finite number of generators, they constitute coordinates on its spectrum (see [28], Ch. III, Theorem 6). We shall use an adequate analog of this fact for non-commutative $C^*$-algebras whose generators are self-adjoint operators with simple spectrum. Namely, with every $\widehat\pi\in\widehat{\mathfrak{E}_{\Sigma}^T}$ we associate the set $\{\widehat\pi|_{\mathfrak{E}_{\gamma}^T}\mid \gamma\in\Sigma\}$ of its restrictions to partial algebras. Each element of this set is a representation of the form (5.4), already provided with $\gamma$-coordinates $t^1_{\gamma},t^2_{\gamma},\dots$ . The correspondence
$$
\begin{equation}
\widehat{\mathfrak{E}_{\Sigma}^T}\ni\widehat\pi \to \bigl\{\{t^1_{\gamma},t^2_{\gamma},\dots\}\bigm| \gamma\in\Sigma\bigr\},\qquad t^k_{\gamma}=t^k_{\gamma}(\widehat\pi),
\end{equation}
\tag{5.5}
$$
provides the required coordinates on the spectrum. We now clarify some details. The correspondence (5.5) is not injective in general, but its restriction to $\operatorname{int}\widehat{\mathfrak{E}_{\Sigma}^T}$ is injective. Since the set $\widehat{\mathfrak{E}_{\Sigma}^T}\setminus\operatorname{int}\widehat{\mathfrak{E}_{\Sigma}^T}$ of all points in the clusters is finite (so, almost all points of the spectrum are interior), the term coordinates seems motivated. If $\widehat\pi\in S_l\subset\mathscr{S}_l$ (see (5.3)), then $\mathbf U_*\widehat\pi$ is an interior point of the spectrum $\widehat{\mathbf U_*\mathfrak{E}_{\Sigma}^T}$ and, by the first representation in (5.1), it corresponds to a certain value of the parameter $r\in (0,\varepsilon_l)$. Then we denote the point $\widehat\pi$ by $\widehat\pi_r$. We shall now find its $\gamma$-coordinates. Suppose that $\varphi\in C(\sigma_{\mathrm{ac}}( E^T_\gamma))$. Among the equivalent representations constituting $\widehat\pi_r$, we choose a representation $\pi_r$ such that
$$
\begin{equation*}
({\mathbf U_*}\pi_r)({\mathbf U}E^T_\gamma)=\sum_{k=1}^{n_{\gamma l}}\tau_{\gamma l}^k(r) P_{\gamma l}^k
\end{equation*}
\notag
$$
(see (5.1)). For this $\pi_r$ we have the following relations:
$$
\begin{equation}
\begin{aligned} \, \notag &\pi_r\bigl(\varphi( E^T_\gamma)\bigr) \stackrel{(4.1)}{=}(\mathbf U_*\pi_r)\bigl(\mathbf U \varphi( E^T_\gamma)\bigr)=(\mathbf U_*\pi_r)\bigl(\varphi(\mathbf UE^T_\gamma)\bigr) \\ \notag &\stackrel{(5.1)}{=}(\mathbf U_*\pi_r)\biggl(\varphi\biggl(\bigoplus\sum_{l=1}^\mathcal L\sum_{k=1}^{n_{\gamma l}}\tau_{\gamma l}^k P_{\gamma l}^k\biggr)\biggr) \\ \notag &\,\,= (\mathbf U_*\pi_r)\biggl(\bigoplus\sum_{l=1}^\mathcal L\sum_{k=1}^{n_{\gamma l}}(\varphi\circ \tau_{\gamma l}^k) P_{\gamma l}^k\biggr) \\ &\,\, =\sum_{k=1}^{n_{\gamma l}}(\varphi\circ\tau_{\gamma l}^k)(r) P_{\gamma l}^k= \sum_{k=1}^{n_{\gamma l}}\varphi(t_{\gamma l}^k) P_{\gamma l}^k =\sum_{k=1}^{n_{\gamma l}}\delta_{t^k_{\gamma l}}(\varphi) P_{\gamma l}^k, \end{aligned}
\end{equation}
\tag{5.6}
$$
where
$$
\begin{equation}
t_{\gamma l}^k = t_{\gamma l}^k(\widehat\pi_r) :=\tau_{\gamma l}^k(r), \qquad r\in(0,\varepsilon_l),\quad k=1,\dots,n_{\gamma l}.
\end{equation}
\tag{5.7}
$$
Comparing the beginning and the end of this calculation, we conclude that the representation $\pi_r|_{\mathfrak{E}^T_\gamma}$ is equivalent to the reduced representation $\bigoplus\sum_{k=1}^{n_{\gamma l}}\delta_{t^k_{\gamma l}}$ of the algebra $\mathfrak{E}^T_\gamma$ and the numbers $\tau_{\gamma l}^k(r)$ are the $\gamma$-coordinates of $\widehat\pi_r\in S_l\subset\operatorname{int}\widehat{\mathfrak{E}^T_\Sigma}$. They are well defined because choosing another $\pi\ne \pi_r$, $\pi\in\widehat\pi_r$, will only result in replacing the projectors $P^k_{\gamma l}$ by unitary equivalent projectors in (5.6). When $\varphi(t)=t$, the relation (5.6) takes the form
$$
\begin{equation}
\pi_r( E^T_\gamma)=\sum_{k=1}^{n_{\gamma l}}\tau^k_{\gamma l}(r)\,P_{\gamma l}^k,\qquad r\in (0,\varepsilon_l),\quad\gamma\in\Sigma,
\end{equation}
\tag{5.8}
$$
which will be used below. We see from (5.7) that when the point $\widehat\pi$ varies in $S_l$, its coordinates $t_{\gamma l}^k(\widehat\pi)$ sweep out the time cells $\psi_{\gamma l}^k\subset\sigma_{\mathrm{ac}}( E^T_\gamma)$ and we have
$$
\begin{equation}
\psi_{\gamma l}^k=\{t_{\gamma l}^k(\widehat\pi)\mid \widehat\pi\in S_l\},\qquad k=1,\dots,n_{\gamma l}, \quad l=1,\dots,\mathcal L,\quad \gamma\in\Sigma.
\end{equation}
\tag{5.9}
$$
This indicates that these cells are invariant and, therefore, so is the partition (5.2). We mean that they are uniquely determined by the eikonal algebra itself or, more precisely, by the structure (5.3) of its spectrum. The cell lengths $\varepsilon_l$, $l=1,\dots,\mathcal L$, are numerical invariants of the algebra ${\mathfrak{E}^T_\Sigma}$. When $k$ is fixed, the correspondence $\operatorname{int}\psi^k_{\gamma l}\ni t^k_{\gamma l}(\widehat\pi)\leftarrow\widehat\pi\in S_l$ is bijective and determines a natural parameterization9 of the interval $\mathscr{S}_l$. For its interior points, we put
$$
\begin{equation}
r(\widehat\pi):=|t^k_{\gamma l}(\widehat\pi)-\tau^k_{\gamma l}(0)|\in(0,\varepsilon_l),\qquad\widehat\pi\in S_l,
\end{equation}
\tag{5.10}
$$
where $\tau^k_{\gamma l}(0)$ is the left end of the cell $\psi^k_{\gamma l}$. Then we extend the parameterization to the “ends” $\mathscr{S}_l\setminus S_l$ by continuity, putting $r=0$ and $r=\varepsilon_l$ respectively. It follows from (5.7) that $r(\widehat{\pi}_r)=r$ for the interior points. Having chosen the parameterization in the way described above, we obviously parameterize the remaining cells $\psi^{k'}_{\gamma l}$, $k'\ne k$, and define functions $\tau^k_{\gamma}(r)$, $r\in(0,\varepsilon_l)$, $k=1,\dots,n_{\gamma l}$, according to (5.7). They are also invariants of the eikonal algebra. 5.3. The transformationThe canonical form (5.1) was obtained by “reformatting” the parametric representation (4.8). We now show how to arrive at it by starting from the original algebra $\mathfrak{E}^T_\Sigma$ and using its invariants. So we have the eikonals $E^T_\gamma$, $\gamma\in\Sigma$, and the algebra $\mathfrak{E}^T_\Sigma$ generated by them. Step 1. Find the spectrum $\widehat{ \mathfrak{E}^T_\Sigma}$, equip it with the (Jacobson) topology and distinguish its closed intervals and their components according to (5.3). Step 2. Find the spectra $\sigma_{\mathrm{ac}}( E^T_\gamma)$ and introduce the $\gamma$-coordinates on $\operatorname{int}\widehat{\mathfrak{E}^T_\Sigma}$. Determine the cells $\psi^k_{\gamma l}$ by (5.9). Parameterize the closed intervals by (5.10). Step 3. For each $l$, choose representations $\pi_r\colon \widehat{\mathfrak{E}^T_\Sigma}\to\mathbb M^{\varkappa_l}$, $\pi_r\,{\in}\,\widehat\pi_r\,{\in}\, S_l$, $r\in(0,\varepsilon_l)$, such that the $\mathbb M^{\varkappa_l}$-valued functions $\pi_r(e)$ are continuous with respect to $r\in(0,\varepsilon_l)$ for all $e\in\mathfrak{E}^T_\Sigma$ and extend them by continuity10 to $[0,\varepsilon_l]$. Step 4. Determine the eigenvalues $\tau^k_{\gamma l}(r)$ and eigenprojectors $P^k_{\gamma l}$ of the matrices $\pi_r(E^T_\gamma)$ (see (5.8)). Using them, define a map
$$
\begin{equation*}
\mathbf{U}\colon E^T_\gamma \mapsto \sum_{k=1}^{n_{\gamma l}}\tau^k_{\gamma l}(\,{\cdot}\,)P_{\gamma l}^k\in \dot C([0,\varepsilon_l];\mathbb M^{\varkappa_l}),\qquad l=1,\dots,\mathcal L,\quad \gamma\in\Sigma,
\end{equation*}
\notag
$$
on the generators (eikonals) and extend it to an algebra isomorphism ${\mathbf U}\colon \mathfrak{E}^T_\Sigma\to\bigoplus_{l=1}^\mathcal L \dot C([0,\varepsilon_l];\mathbb M^{\varkappa_l})$. The map $\mathbf U$ transforms $\mathfrak{E}^T_\Sigma$ to a canonical form (or rather one of its versions: see the comment after Theorem 2). All our results concern the shifted eikonals $\dot E^T_\gamma$ (see Convention 2). Their reformulation for the original eikonals $E^T_\gamma=\dot E^T_\gamma-P^T_\gamma$ is obvious. It suffices to replace the functions $\tau^k_{\gamma l}$ in (5.1) by $\tau^k_{\gamma l}-1$. CommentsThe map ${\mathbf V}\colon \mathfrak{E}^T_\Sigma\to\bigoplus_{l=1}^\mathcal L \dot C(\mathscr{S}_l;\mathbb M^{\varkappa_l})$, which is defined on the generators by the formula
$$
\begin{equation*}
\bigl(\mathbf{V}(E^T_\gamma)\bigr)(\widehat\pi):= \sum_{k=1}^{n_{\gamma l}} \tau^k_{\gamma l}(r(\widehat\pi)) P_{\gamma l}^k,\qquad \widehat\pi\in\mathscr{S}_l,\quad l=1,\dots,\mathcal L,\quad\gamma\in\Sigma,
\end{equation*}
\notag
$$
realizes the elements of the algebra $\mathfrak{E}^T_\Sigma$ as matrix-valued functions on its spectrum and thus provides an invariant functional model of the eikonal algebra. One can arrive at this model with all its attributes (the cells $\psi^k_{\gamma l}$, the functions $\tau^k_{\gamma l}$, the lengths $\varepsilon_l$, the dimensions $\varkappa_l$) starting with any isomorphic copy of the algebra $\mathfrak{E}^T_\Sigma$ and making Steps 1–4. This is important for the inverse problem since its data determine such a copy. The question is to what extent do these attributes determine the structure of the graph $\Omega$. The following observation may be useful in the inverse problem. Define a relation between points of the spectrum $\widehat{\mathfrak{E}^T_\Sigma}$ by declaring that $\widehat\pi\stackrel{\gamma}{\sim_0}\widehat\pi'$ if some $\gamma$-coordinates of these points coincide. i.e. $\tau^k_{\gamma l}(\widehat\pi)=\tau^{k'}_{\gamma l'}(\widehat\pi')$. Then define an equivalence by writing $\widehat\pi\sim\widehat\pi'$ if there are points $\widehat\pi_1,\dots,\widehat\pi_p\in\widehat{\mathfrak{E}^T_\Sigma}$ and vertices $\gamma_1,\dots, \gamma_{p+1}\in\Sigma$ such that $\widehat\pi\stackrel{\gamma_1}\sim_0\widehat\pi_1\stackrel{\gamma_2}\sim_0\cdots \stackrel{\gamma_p}\sim_0\widehat\pi_p\stackrel{\gamma_{p+1}}{\sim_0}\widehat\pi'$. We can show that taking the quotient of the spectrum with respect to the relation $\widehat\pi\sim\widehat\pi'$ identifies only the points lying in the clusters. Moreover, this quotient of $\widehat{\mathfrak{E}^T_\Sigma}$ is homeomorphic to a graph. Examples show that the resulting graph is homeomorphic to the quotient of the wave-filled region $\Omega^T_\Sigma$ by a certain relation which has a simple geometric meaning. In known examples [6], [8], the clusters arise only when an interior vertex of $\Omega$ is overlapped by waves coming from at least two boundary vertices. We conjecture that this is a general fact. It is also interesting whether the presence of cycles in $\Omega^T_\Sigma$ can be characterized in terms of the algebra $\mathfrak{E}^T_\Sigma$ (see [6]). This question is open.
Citation in format AMSBIB
\Bibitem{BelKap22} |
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