| Sbornik: Mathematics |
|
|
|
|
|
The spectrum of the I. V. Baibulov, O. V. Sarafanov St. Petersburg University, St. Petersburg, Russia
Bibliographic databases:
     § 1. IntroductionIn this work we study a $C^*$-algebra generated by singular integral operators with semi-almost periodic coefficients in the space $L_2(\mathbb{R})$ (see the precise definitions in § 2). The goal is to describe the primitive spectrum of this algebra, that is, to list all of its primitive ideals and describe the Jacobson topology. The $C^*$-algebra of multidimensional singular integral operators with smooth coefficients that have finite limits at infinity was considered in [1]. It was shown that the quotient algebra by the ideal of compact operators is commutative, and the set of its maximal ideals was described. The spectrum (set of classes of unitary equivalence of irreducible representations) of the $C^*$-algebra of singular integral operators on a complex contour with discontinuities of the first kind in coefficients was studied in [2] and [3]. In the works mentioned above the algebras are GCR (postliminal, type I), so their spectrum is homeomorphic to their primitive spectrum. The algebra considered in this work is not of this type; we do not aim to list all of its irreducible representations. Various classes of operators with semi-almost periodic coefficients (pseudodifferential operators, Toeplitz operators, Hankel operators, Wiener–Hopf operators) have been studied previously in terms of the Fredholm theory (see, for example, [4] and the references there). The algebra considered in our paper was introduced in [5], where a criterion for the Fredholm property of elements of this algebra was established and formulae for the index of certain Fredholm operators were obtained; however, the spectrum of the algebra was not investigated. Let us briefly outline the content of the paper. Section 1 is the introduction. In § 2 we introduce the $C^*$-algebra under consideration $\mathfrak{T}$ and apply to it the localization principle from [6] and [7]. The spectrum of the initial algebra is the union of the spectra of the ‘local’ algebras, which are obtained, roughly speaking, by ‘freezing’ the coefficients at a fixed point on the real line. Only the local algebra $\mathfrak{T}_\infty$ corresponding to the infinity requires a separate consideration. It turns out that $\mathfrak{T}_\infty$ is isomorphic to the crossed product of a certain algebra $\mathcal{B}_\infty$ (considered in [8]) and the group $\mathbb{R}_d$ of real numbers with discrete topology: see Theorem 1. The proof of Theorem 1 begins in § 3. In § 4 we provide the necessary information about the spectrum of the algebra $\mathcal{B}_\infty$ without a proof. In § 5 it is shown that the action $\alpha$ of the group $\mathbb{R}_d$ on the algebra $\mathcal{B}_\infty$ is topologically free, and the proof of Theorem 1 is completed. In § 6 we describe the space of quasi-orbits of the $C^*$-dynamical system $(\mathcal{B}_\infty,\mathbb{R}_d,\alpha)$ and establish that this space is almost Hausdorff. To describe the spectrum of $\mathfrak{T}_\infty= \mathcal{B}_\infty \rtimes_{\alpha} \mathbb{R}_d$ we use the localization principle on quasi-orbits, which is proved in § 7. Statements similar to this localization principle were formulated in [9], formula 4.3, and [10], Corollary 7.27, under excessively restrictive assumptions. Specifically, in [9], the algebra appearing in the crossed product is required to be separable, and the group to be second countable. In [10], as applied to our situation, one of the requirements comes down to each orbit being homeomorphic to the group $\mathbb{R}_d$ or consisting of a single point. We prove the localization principle on quasi-orbits by assuming only that the group is discrete and the space of quasi-orbits of the dynamical system is almost Hausdorff. In § 8 the primitive spectrum of the local algebra $\mathfrak{T}_\infty$ is described, including the Jacobson topology. Each quasi-orbit of the $C^*$-dynamical system $(\mathcal{B}_\infty, \mathbb{R}_d, \alpha)$ is associated with an auxiliary $C^*$-algebra. Its spectrum is identified with a part of the spectrum of the algebra $\mathfrak{T}_\infty$. The Jacobson topology on the spectrum of the auxiliary algebra is transferred to the corresponding part of the spectrum of the algebra $\mathfrak{T}_\infty$. From the localization principle it follows that these parts do not intersect; after an appropriate gluing procedure the Jacobson topology on the spectrum of the algebra $\mathfrak{T}_\infty$ is obtained. The primitive ideals of the algebra $\mathfrak{T}$ are listed in § 9, and the Jacobson topology on $\operatorname{Prim} \mathfrak{T}$ is described in § 10. Throughout the paper, by an algebra we mean a $C^*$-algebra, by an algebra homomorphism we mean a $*$-homomorphism, and by an ideal we mean a closed two-sided ideal. § 2. The algebra of singular integral operators. Local algebrasLet $C_b(\mathbb{R})$ be the algebra of continuous functions on the real line with the pointwise operations, the complex conjugation as an involution, and the norm and let $C_0(\mathbb{R})$ be the subalgebra in $C_b(\mathbb{R})$ consisting of all functions vanishing at infinity. We denote the one-point compactification of the real line by $\dot{\mathbb{R}}$ and the two-point compactification by $\overline{\mathbb{R}}$. Let $C(\dot{\mathbb{R}})$ and $C(\overline{\mathbb{R}})$ be the subalgebras in $C_b(\mathbb{R})$ that consist of all functions admitting a continuous extension to $\dot{\mathbb{R}}$ and $\overline{\mathbb{R}}$, respectively.We denote by $\operatorname{AP}(\mathbb{R})$ the subalgebra of $C_b(\mathbb{R})$ generated by the linear combinations of the form
$$
\begin{equation*}
\sum_{n=1}^N c_n \exp(ik_nx), \qquad k_n\in\mathbb{R};
\end{equation*}
\notag
$$
elements of $\operatorname{AP}(\mathbb{R})$ are called almost-periodic functions (see [11] and [12]). Let $\operatorname{SAP}(\mathbb{R})$ denote the subalgebra of $C_b(\mathbb{R})$ spanned by the algebras $\operatorname{AP}(\mathbb{R})$ and $C(\overline{\mathbb{R}})$; its elements are called semi-almost periodic functions. Any semi-almost periodic function can be represented in the form where $f_\pm \in \operatorname{AP}(\mathbb{R})$, $f_0\in C_0(\mathbb{R})$ and $\chi_\pm \in C(\overline{\mathbb{R}})$ with $\chi_+(+\infty)=1$, ${\chi_+(-\infty)=0}$, and $\chi_-=1-\chi_+$. It is clear that $f_\pm$ do not depend on the choice of $\chi_+$. Let $\mathfrak{T}$ denote the algebra generated in $L_2(\mathbb{R})$ by the singular integral operators of the form ${A=aI+bS}$, where
$$
\begin{equation}
(Su)(x)=\frac{1}{\pi i} \operatorname{p.v.} \int_\mathbb{R} \frac{u(y)\,dy}{y-x},
\end{equation}
\tag{2.2}
$$
and $a$, $b \in \operatorname{SAP}(\mathbb{R})$. We will reduce the problem of the description of the spectrum of the algebra $\mathfrak{T}$ to the description of the spectra of simpler ‘local’ algebras. To do this we apply the ‘localization principle’ from [6]; also see [7]. The localization principle in various forms was used by Simonenko [13], [14], Douglas [15] and Dynin [16]. Let $\mathcal{A}$ be a $C^*$-algebra, $\mathcal{C}$ a commutative subalgebra of $\mathcal{A}$, $\widehat{\mathcal{C}}$ the spectrum of $\mathcal{C}$ and $\widehat{c}(\,\cdot\,)$ the Gelfand transform of the element $c \in \mathcal{C}$. For each point $x\in \widehat{\mathcal{C}}$, let $\mathcal{I}_x=\{c\in \mathcal{C} \colon \widehat{c}(x)=0 \}$; we denote by $\mathcal{J}_x$ the ideal of $\mathcal{A}$ generated by $\mathcal{I}_x$. The quotient algebras $\mathcal{A}_x=\mathcal{A}/\mathcal{J}_x$ are called local algebras, and the transition from $\mathcal{A}$ to $\mathcal{A}_x$ is called localization. The spectrum $\widehat{\mathcal{A}_x}$ is naturally identified with a subset of $\widehat{\mathcal{A}}$. Therefore, the following inclusion holds:
$$
\begin{equation*}
\widehat{\mathcal{A}} \supset \bigcup_{x \in \widehat{C}} \widehat{\mathcal{A}_x}.
\end{equation*}
\notag
$$
Proposition 1 (localization principle). Let $J$ be an ideal in $\mathcal{A}$. Suppose that Remark 1. From the proof of the localization principle it follows that if $x \neq y$, then $\widehat{\mathcal{A}_x} \cap \widehat{\mathcal{A}_y}=\varnothing$. We assume the commutative subalgebra $\mathcal{C}$ to be the algebra of multiplication operators by functions in $C(\dot{\mathbb{R}})$. The spectrum of this algebra is homeomorphic to $\dot{\mathbb{R}}$. Note that the algebra $\mathfrak{T}$ contains the subalgebra of singular integral operators with coefficients in $C(\dot{\mathbb{R}})$. This subalgebra is irreducible and contains the ideal of compact operators $\mathrm{KL}_2(\mathbb{R})$ (see [8], Propositions 4.3 and 4.4). Consequently, $\mathfrak{T}$ is also irreducible and contains $\mathrm{KL}_2(\mathbb{R})$. In particular, this means that the identity representation of the algebra $\mathfrak{T}$ is irreducible. Set $J=\mathrm{KL}_2(\mathbb{R})$. Proposition 2. The following equality holds: where $[\operatorname{Id}]$ is the class of the identity representation. Proof. Let us check that the localization principle (Proposition 1) is applicable to $\mathcal{C}=C(\dot{\mathbb{R}})$ (in our notation we sometimes do not distinguish between functions and the operators of multiplication by these functions) and $J=\mathrm{KL}_2(\mathbb{R})$. Condition (i) is satisfied since both the algebras $\mathfrak{T}$ and $C(\dot{\mathbb{R}})$ contain the identity element.
To verify condition (ii) note that the commutator $[c,A]$ is compact for $c\in C_c^\infty(\mathbb{R})$ and $A\in \mathfrak{T}$. Indeed, any element $A\in\mathfrak{T}$ can be approximated by elements of the form where $a_{lk},b_{lk} \in \operatorname{SAP}(\mathbb{R})$. It remains to take into account that $[c, S] \in \mathrm{KL}_2(\mathbb{R})$ if ${c\in C_c^\infty(\mathbb{R})}$.Let $x_1,x_2\in\dot{\mathbb{R}}$ be two distinct points, and let $c_1$ and $c_2$ be smooth functions such that $c_j(x_j)\neq0$ and $c_1c_2=0$. If $x_j\neq\infty$, then we choose the corresponding function $c_j$ in $C_c^\infty(\mathbb{R})$. Suppose, for example, that $x_1\neq\infty$; then for any $A\in\mathfrak{T}$ we have
$$
\begin{equation*}
c_1 A c_2 =A c_1 c_2+[c_1,A]c_2=[c_1,A]c_2 \in \mathrm{KL}_2(\mathbb{R}).
\end{equation*}
\notag
$$
If $x_1=\infty$, then $x_2\neq\infty$ and
$$
\begin{equation*}
c_1 A c_2=c_1 c_2 A+c_1[A,c_2]=c_1[A,c_2] \in \mathrm{KL}_2(\mathbb{R}).
\end{equation*}
\notag
$$
Hence condition (ii) is fulfilled.
Finally, condition (iii) can easily be verified, since each irreducible representation of $\mathrm{KL}_2$ is equivalent to the identity representation. We fix an arbitrary point $x\in \dot{\mathbb{R}}$ and let $c\in C_c^\infty(\mathbb{R})$ be a nonzero function satisfying $c(x)=0$. We choose nonzero $\phi,\psi \in C_c^\infty(\mathbb{R})$ such that $\psi c= c$ and define the operator $B_{\phi,\psi}$ by Then $B_{\phi,\psi}\in \mathrm{KL}_2(\mathbb{R})$, and the operator $B_{\phi,\psi}c$ is nonzero.The proposition is proved. Let us describe the spectrum of the local algebras $\mathfrak{T}_{x}$ for $x\neq\infty$; this result is essentially contained in [2] and [3]. For $x_0\in\mathbb{R}$ let $[A]_{x_0} \in \mathfrak{T}_{x_0}$ be the equivalence class of $A\in \mathfrak{T} $. Clearly, the operator $a(x_0)I+b(x_0)S$ is a representative of the class $[aI+bS]_{x_0}$. Thus, the algebra $\mathfrak{T}_{x_0}$ is generated by the element $[S]_{x_0}$; in particular, $\mathfrak{T}_{x_0}$ is commutative. As the spectrum of $S$ consists of the two points $\pm 1$, the set of characters of $\mathfrak{T}_{x_0}$ consists of two elements $\pi(x_0,+)$ and $\pi(x_0,-)$; their action on the generators is given by
$$
\begin{equation}
\pi(x_0,\pm)\colon[aI+bS]_{x_0} \mapsto a(x_0) \mp b(x_0).
\end{equation}
\tag{2.4}
$$
The map $\pi(x_0,+)\oplus\pi(x_0,-)\colon\mathfrak{T}_{x_0}\mapsto\mathbb{C}\oplus\mathbb{C}$ is the Gelfand isomorphism.
§ 3. Crossed product structure in $\mathfrak{T}_\infty$We provide a definition of a crossed product $\mathcal{A}\rtimes_{\alpha} G$ in the special case of discrete group $G$ (for the general case, see [17]). Suppose that $\mathcal{A}$ is some $C^*$-algebra, $G$ is a discrete group, and $\alpha\colon t\mapsto \alpha_t$ is a fixed homomorphism from $G$ to the automorphism group of $\mathcal{A}$. That is, the triple $\left(\mathcal{A},G,\alpha\right)$ is a $C^*$-dynamical system; in what follows, by a dynamical system we always mean such a $C^*$-dynamical system. The homomorphism $\alpha$ is called the action of the group $G$ on the algebra $\mathcal{A}$. We denote by $C_c(G,\mathcal{A})$ the set of functions from $G$ to $\mathcal{A}$ that take nonzero values only at finitely many points. The multiplication and involution operations are defined by and It is known that the set $C_c(G,\mathcal{A})$ equipped with these operations forms a $*$-algebra.Let $\pi\colon\mathcal{A}\to B(\mathcal{H})$ be a non-degenerate representation of $\mathcal{A}$ and $U\colon G\to B(\mathcal{H})$ be a unitary representation of $G$ on the same Hilbert space $\mathcal{H}$. Suppose that, for any $t \in G$ and $a \in \mathcal{A}$, the following relation holds: then the pair $(\pi,U)$ is called the covariant representation of the dynamical system $(\mathcal{A},G,\alpha)$. Every covariant representation $(\pi,U)$ of $(\mathcal{A},G,\alpha)$ is associated with the representation $\pi\rtimes U$ of the $*$-algebra $C_c(G,\mathcal{A})$:
$$
\begin{equation}
(\pi \rtimes U)(f)=\sum_{t\in G} \pi(f(t)) U(t), \qquad f\in C_c(G,\mathcal{A}).
\end{equation}
\tag{3.1}
$$
The completion of $C_c(G,\mathcal{A})$ with respect to the norm
$$
\begin{equation}
\|f\|=\sup \bigl\{ \|\pi\rtimes U (f)\|\colon (\pi,U) \text{ is a covariant representation of } (\mathcal{A},G,\alpha)\bigr\}
\end{equation}
\tag{3.2}
$$
is a $C^*$-algebra (see [17], Lemma 2.27). This algebra is called the crossed product of $\mathcal{A}$ and $G$ and is denoted by $\mathcal{A}\rtimes_{\alpha}G$. In this work we sometimes use the notion of so-called reduced crossed product (denoted by $\mathcal{A}\rtimes_{r,\alpha}G$), see [17], Definition 7.7. The precise definition is not provided here, as it is not needed for our purposes. We only make use of certain results related to the reduced crossed product, which are, in general, invalid for the ‘full’ product $\mathcal{A}\rtimes_{\alpha} G$. We rely on the fact that, throughout this work, the group $G$ is abelian (hence amenable), and therefore $\mathcal{A}\rtimes_{r,\alpha} G \cong \mathcal{A}\rtimes_{\alpha} G$ (see [17], Theorem 7.13). We now proceed to study the algebra $\mathfrak{T}_\infty=\mathfrak{T} / \mathcal{J}_\infty$. We let $G$ be the group $\mathbb{R}_d$ of real numbers with discrete topology. For each $k\in \mathbb{R}_d$ we define $U(k)=[e_k]_\infty$, where $e_k \in \mathfrak{T}$ is the multiplication operator by the function $e_k(x)=e^{-ikx}$, and $[e_k]_\infty$ is its class in $\mathfrak{T}_\infty$. Clearly, $U\colon\mathbb{R}_d\to \mathfrak{T}_\infty$ is a unitary homomorphism. The action of the group $\mathbb{R}_d$ on $\mathfrak{T}_\infty$ is defined by the equality
$$
\begin{equation}
\alpha_k\colon [A]_\infty \mapsto [e_kAe_k^*]_\infty=U(k)[A]_\infty U(k)^*.
\end{equation}
\tag{3.3}
$$
Now we introduce a subalgebra $\mathcal{B}_\infty$ in $\mathfrak{T}_\infty$ playing the role of $\mathcal{A}$ in the crossed product. We denote by $\mathcal{V}$ the subalgebra in $\operatorname{BL}_2(\mathbb{R})$ generated by the operators of the form $F^{-1}h_kF$, where $h_k(\xi)=\operatorname{sgn}(\xi+k)$, $k \in \mathbb{R}$. The inclusion $ \mathcal{V} \subset \mathfrak{T}$ holds, since where $S$ is the operator in (2.2). Let $\mathcal{B}$ be the subalgebra in $\mathfrak{T}$ spanned by $\mathcal{V}$ and $C(\overline{\mathbb{R}})$, and let $\mathcal{B}_\infty$ be its image under the projection $\mathfrak{T} \to \mathfrak{T}_\infty$. Since $e_k\mathcal{V}e_k^*\subset \mathcal{V}$ and $e_kC(\overline{\mathbb{R}})e_k^* \subset C(\overline{\mathbb{R}})$, we have $e_k\mathcal{B}e_k^* \subset \mathcal{B}$ and $\alpha_k(\mathcal{B}_\infty) \subset \mathcal{B}_\infty$. We denote the restriction of $\alpha_k$ to $\mathcal{B}_\infty$ by the same symbol $\alpha_k$. It is straightforward to see that the action of $\alpha_k$ on the generators of $\mathcal{B}_\infty$ is given by
$$
\begin{equation}
\alpha_k([a]_\infty)=[a]_\infty, \qquad a \in C(\overline{\mathbb{R}}),
\end{equation}
\tag{3.4}
$$
and
$$
\begin{equation}
\alpha_k([F^{-1}h_lF]_\infty)=[F^{-1}h_{l+k}F]_\infty, \qquad h_l(\xi)=\operatorname{sgn}(\xi+l).
\end{equation}
\tag{3.5}
$$
The triple $(\mathcal{B}_\infty,\mathbb{R}_d,\alpha)$ is a dynamical system. Theorem 1. The algebras $\mathcal{B}_\infty \rtimes_\alpha \mathbb{R}_d$ and $\mathfrak{T}_\infty$ are isomorphic. Proof. Consider the map
$$
\begin{equation}
\Phi_U\colon C_c(\mathbb{R}_d,\mathcal{B}_\infty) \to \mathfrak{T}_\infty\colon f \mapsto \sum_{k\in \mathbb{R}_d} f(k) U(k).
\end{equation}
\tag{3.6}
$$
Clearly, $\Phi_U$ is a $*$-homomorphism. Let $\widetilde{\pi}\colon \mathfrak{T}_\infty \to B(\mathcal{H})$ be some faithful representation of $\mathfrak{T}_\infty$ in the Hilbert space $\mathcal{H}$, and let $\pi$ be its restriction to $\mathcal{B}_\infty$. Then the pair $(\pi, \widetilde{\pi} \circ U)$ is a covariant representation of the dynamical system $(\mathcal{B}_\infty, \mathbb{R}_d, \alpha)$, and $\widetilde{\pi}\circ \Phi_U=\pi\rtimes (\widetilde{\pi}\circ U)$ is the associated representation of the $*$-algebra $C_c(\mathbb{R}_d,\mathcal{B}_\infty)$. The faithfulness of $\widetilde{\pi}$ and the definition of the norm (3.2) imply that Therefore, $\Phi_U$ is bounded and extends continuously to a $*$-homomorphism from $\mathcal{B}_\infty \rtimes_{\alpha}\mathbb{R}_d$ to $\mathfrak{T}_\infty$. We denote this extension by $\Phi_U$ as well.
Let us show that the homomorphism $\Phi_U$ is surjective. Since the image of any $*$-homomorphism of $C^*$-algebras is closed, it is sufficient to prove that the set $\Phi_U(C_c(\mathbb{R}_d, \mathcal{B}_\infty))$ is dense in $\mathfrak{T}_\infty$. In other words, we only need to show that each element of the algebra $\mathfrak{T}_\infty$ can be approximated by finite sums of the form where $k_j \in \mathbb{R}_d$ and $ [b_j]_\infty \in \mathcal{B}_\infty$. The algebra $\mathfrak{T}_\infty$ is generated by the elements $[a]_\infty$ and $[S]_\infty$, where $a \in \operatorname{SAP}(\mathbb{R})$. Any element $a \in \operatorname{SAP}(\mathbb{R})$ can be approximated by finite sums of the form where $b_j\in C(\overline{\mathbb{R}})$. Therefore, $[a]_\infty$ can be approximated by finite sums of the form It remains to note that $[S]_\infty \in \mathcal{B}_\infty$. To complete the proof we need to verify that the ideal $\ker \Phi_U$ is trivial. This is done in § 5.§ 4. The spectrum of the algebra $\mathcal{B}_\infty$In this section we collect the necessary information on the spectrum of the algebra $\mathcal{B}_\infty$. The required results could be obtained directly by applying the localization principle from Proposition 1 to $\mathcal{B}_\infty$. Instead, we show that $\mathcal{B}_\infty$ is isomorphic to an algebra that was studied in [8]. Consider the $C^*$-algebra $\mathfrak{S}$ generated by the operator $S$ and the operators of multiplication by functions in $C(\overline{\mathbb{R}})$ and $h_k$ (see the notation after (3.3)). We denote by $\mathcal{J}_{\text{fin}}$ the ideal in $\mathfrak{S}$ generated by the commutator $[S,\chi]$, where $\chi $ is a continuous function on $\overline{\mathbb{R}}$ such that $\chi(+\infty)=1$ and $\chi(-\infty)=-1$. Lemma 1. The algebras $\mathcal{B}_\infty$ and $\mathfrak{S}/\mathcal{J}_{\text{fin}}$ are isomorphic. Proof. Let $\widetilde{\mathcal{B}}$ be the $C^*$-algebra generated by $\mathcal{B}$ and the operators of multiplication by $h_k$. We denote by $\widetilde{\mathcal{J}}_\infty$ and $\widetilde{\mathcal{J}}^\infty$ the ideals in $\widetilde{\mathcal{B}}$ generated by the subsets $C_0(\mathbb{R})$ and $F^{-1}C_0(\mathbb{R})F$, respectively. The equalities $\widetilde{\mathcal{B}}=\mathfrak{S}+\widetilde{\mathcal{J}}^\infty$ and $\mathcal{J}_{\text{fin}}=\mathfrak{S}\cap \widetilde{\mathcal{J}}^\infty$ hold (see [8], Lemmas 5.6 and 5.7). Therefore, the inclusion $\mathfrak{S} \subset \widetilde{\mathcal{B}}$ induces an isomorphism Clearly, $F\widetilde{\mathcal{B}}F^{-1}= \widetilde{\mathcal{B}}$ and $F\widetilde{\mathcal{J}}^\infty F^{-1}=\widetilde{\mathcal{J}}_\infty$. Consequently, the Fourier transform gives rise to an isomorphism between $\widetilde{\mathcal{B}}/\widetilde{\mathcal{J}}_\infty$ and $F\mathfrak{S}F^{-1}/(F\mathcal{J}_{\text{fin}} F^{-1}) \cong \mathfrak{S}/\mathcal{J}_{\text{fin}}$.
Recall that $\mathcal{B}_\infty$ is the image of $\mathcal{B}$ under the quotient map $\mathfrak{T} \to \mathfrak{T}/\mathcal{J}_\infty$, that is, $\mathcal{B}_\infty =\mathcal{B}/(\mathcal{B}\cap \mathcal{J}_\infty)$. First we demonstrate that $\mathcal{B}\cap \mathcal{J}_\infty=\mathcal{B}\cap \widetilde{\mathcal{J}}_\infty$, implying that $\mathcal{B}_\infty=\mathcal{B}/(\mathcal{B}\cap \widetilde{\mathcal{J}}_\infty)$. After that we establish that $\mathcal{B}/(\mathcal{B}\cap \widetilde{\mathcal{J}}_\infty)=\widetilde{\mathcal{B}}/\widetilde{\mathcal{J}}_\infty$. Consider a sequence of functions $\phi_n \in C_0(\mathbb{R})$ such that $\phi_n(x)=1$ for $x \in [-n, n]$. It follows from Lemma 5.4 in [8] that $\phi_n$ is an approximate identity for the ideal $\widetilde{\mathcal{J}}_\infty$. Since $\phi_n \in \mathcal{B}$, this implies that $\phi_n$ is also an approximate identity for the ideal $\mathcal{B}\cap \widetilde{\mathcal{J}}_\infty$ of $\mathcal{B}$. In other words, if $b \in \mathcal{B}$, then
$$
\begin{equation}
b \in \mathcal{B}\cap \widetilde{\mathcal{J}}_\infty \quad\Longleftrightarrow\quad \|(1-\phi_n)b\| \to 0.
\end{equation}
\tag{4.1}
$$
In particular, since $\mathrm{KL}_2(\mathbb{R}) \subset \widetilde{\mathcal{J}}_\infty$, it follows that $\left\| (1-\phi_n)C \right\| \to 0$ for each ${C \in \mathrm{KL}_2(\mathbb{R})}$. Let us show that the sequence $\phi_n$ also serves as an approximate identity for the ideal $\mathcal{B}\cap \mathcal{J}_\infty$. That is, if $b \in \mathcal{B}$, then
$$
\begin{equation}
b \in \mathcal{B}\cap \mathcal{J}_\infty \quad\Longleftrightarrow\quad \|(1-\phi_n)b\| \to 0.
\end{equation}
\tag{4.2}
$$
Indeed, any element in $\mathcal{J}_\infty$ can be approximated by combinations of the form
$$
\begin{equation*}
\sum_{j=1}^K A_{1j}a_j A_{2j}, \qquad A_{ij}\in \mathfrak{T}, \quad a_j \in C_0(\mathbb{R}).
\end{equation*}
\notag
$$
Since $[A_{ij},a_{j}]\in \mathrm{KL}_2$ for $a_{j} \in C_0(\mathbb{R})$, it suffices to consider all possible sums of the form
$$
\begin{equation*}
aA+C, \qquad \text{where}\quad a\in C_0(\mathbb{R}), \quad A\in \mathfrak{T}\quad\text{and} \quad C\in \mathrm{KL}_2(\mathbb{R}).
\end{equation*}
\notag
$$
From this and the equality we obtain $\|(1-\phi_n)b \| \to 0$, as required. Since $\phi_n \in \mathcal{J}_\infty\cap \mathcal{B}$, the reverse implication follows from the closedness of the ideal $\mathcal{J}_\infty \cap \mathcal{B}$. Thus, $\mathcal{B}\cap \mathcal{J}_\infty=\mathcal{B}\cap \widetilde{\mathcal{J}}_\infty$.
To prove the relation $\mathcal{B}/(\mathcal{B}\cap \widetilde{\mathcal{J}}_\infty)=\widetilde{\mathcal{B}}/\widetilde{\mathcal{J}}_\infty$, note that the inclusion $\mathcal{B}\subset \widetilde{\mathcal{B}}$ induces the embedding $\mathcal{B}_\infty \to \widetilde{\mathcal{B}}/\widetilde{\mathcal{J}}_\infty$. In addition to elements of $\mathcal{B}$, the algebra $\widetilde{\mathcal{B}}$ is also generated by the multiplication operators $h_k$. Since $h_k - \chi \in \widetilde{\mathcal{J}}_\infty$ and $\chi \in \mathcal{B}$ ($\chi$ was defined before the lemma), the above embedding is surjective and therefore an isomorphism. The lemma is proved. It follows from the proof of Lemma 1 that the action of the isomorphism ${\mathfrak{S}/\mathcal{J}_{\text{fin}} \cong \mathcal{B}_\infty}$ can, loosely speaking, be reduced to the transformation $a \mapsto F^{-1} a F$ and the inclusion operation. In the process, the class of the operator $S$ is mapped to the class $[-\chi]_\infty$, and the classes of the multiplication operators by $h$ are mapped to the classes $[F^{-1}hF]_\infty$. The irreducible representations of $\mathfrak{S}/\mathrm{KL}_2$ are listed in [8], Theorem 4.13, and the Jacobson topology on the spectrum is given in [8], Theorem 4.16 (in the notation of these theorems, the set $X$ denotes the set of discontinuities of the coefficients, and, in the case of the algebra $\mathfrak{S}$, it coincides with $\dot{\mathbb{R}}$). From Proposition 4.20(2) in [8] for $X=\dot{\mathbb{R}}$ it follows that the spectrum of $\mathfrak{S}/\mathcal{J}_{\text{fin}}$ coincides with the subset of $\widehat{\mathfrak{S}/\mathrm{KL}_2}$ obtained by excluding the representations $\{\pi(\infty,\lambda)\colon \lambda \in \mathbb{R}\}$ (in the notation of [8], Theorem 4.13). For convenience, we replace the two-dimensional representations $\pi(x,\lambda)$ in this list by the unitarily equivalent representations $ U \pi(x,\lambda) U^*$, where
$$
\begin{equation*}
U=2^{1/2} e^{i\pi/4} \begin{pmatrix} i \sqrt{1+\tanh(\pi\lambda)} & -i\sqrt{1-\tanh(\pi\lambda)}\,\,\,\, \\ -i\sqrt{1-\tanh(\pi\lambda)} & \sqrt{1+\tanh(\pi\lambda)} \end{pmatrix},
\end{equation*}
\notag
$$
so that the class $[-\chi]_\infty$ is mapped to the diagonal operator $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} $. Let us summarize the final result. Set $\mathbb{R}_{(+)}=[-\infty,+\infty)$ and $\mathbb{R}_{(-)}=(-\infty,+\infty]$. Let us define the maps $\pi(\phi,\omega,\xi)$ for $\phi=\pm \infty$, $\omega=\pm$, $\xi \in \mathbb{R}_{(\omega)}$ on the generators $[a]_{\infty}$ and $[F^{-1}bF]_{\infty}$ of $\mathcal{B}_\infty$ (where $a \in C(\overline{\mathbb{R}})$, $F^{-1}bF \in \mathcal{V}$) by
$$
\begin{equation}
\pi(\phi,\pm,\xi)([a]_\infty) =\lim_{x\to \phi} a(x),
\end{equation}
\tag{4.3}
$$
$$
\begin{equation}
\pi(\phi, \pm, \xi)([F^{-1}bF]_\infty) =\lim_{\varepsilon\to+0 } b(\xi \pm \varepsilon), \qquad \xi \neq \infty,
\end{equation}
\tag{4.4}
$$
and
$$
\begin{equation}
\begin{aligned} \, \pi(\phi,+,-\infty)([F^{-1}bF]_\infty) &=\lim_{\xi \to - \infty} b(\xi), \\ \pi(\phi, -,+\infty)([F^{-1}bF]_\infty)&=\lim_{\xi \to+ \infty} b(\xi). \end{aligned}
\end{equation}
\tag{4.5}
$$
Note that $\omega$ and $\phi$ take values independently; thus, for each $\xi$ there are, in general, four different maps. We interpret the limit of the function $b$ as $\xi \to -\infty$ as the right limit at the point at infinity, and so $\pi(\phi,+, -\infty)$ is included in the series $\pi(\phi,+, \xi)$ for ${\xi \in \mathbb{R}_{(+)}}$. For a similar reason $\pi(\phi, -, +\infty)$ is included in the series $\pi(\phi, -, \xi)$. For $\xi \in \mathbb{R}$ and $\lambda \in \mathbb{R}$ we introduce the maps $\pi(\xi,\lambda)$ with values in $M_2(\mathbb{C})$ which are defined on the generators of $\mathcal{B}_\infty$ by the formulae
$$
\begin{equation}
\pi(\xi,\lambda)([a]_\infty) =\begin{pmatrix} a(+\infty) & 0 \\ 0 & a(-\infty) \end{pmatrix}
\end{equation}
\tag{4.6}
$$
and
$$
\begin{equation}
\pi(\xi,\lambda)([F^{-1}bF]_\infty)= -\frac{b(\xi+0)-b(\xi-0)}{2}B(\lambda) +\frac{b(\xi+0)+b(\xi-0)}{2}I,
\end{equation}
\tag{4.7}
$$
where
$$
\begin{equation}
B(\lambda)=\begin{pmatrix} -\tanh(\pi\lambda) & -\dfrac{i}{\cosh(\pi\lambda)} \\ \dfrac{i}{\cosh(\pi\lambda)} & \tanh(\pi\lambda) \end{pmatrix}.
\end{equation}
\tag{4.8}
$$
Lemma 2. Each of the maps $\pi(\phi,\omega,\xi)$ and $\pi(\xi,\lambda)$ extends to a representation of $\mathcal{B}_\infty$. The resulting representations are irreducible and pairwise inequivalent. Any irreducible representation of $\mathcal{B}_\infty$ is equivalent to one of the representations $\pi(\xi,\omega,\phi)$ and $\pi(\xi,\lambda)$. We define the set $\Sigma$ parametrizing $\widehat{\mathcal{B}_\infty}$. Consider the disjoint union
$$
\begin{equation}
\Sigma=(\mathbb{R}_d\times \mathbb{R}) \cup \mathbb{R}_{(+)}^+\cup \mathbb{R}_{(-)}^+\cup \mathbb{R}_{(+)}^- \cup \mathbb{R}_{(-)}^-,
\end{equation}
\tag{4.9}
$$
where $\mathbb{R}_{(+)}^\pm$ and $\mathbb{R}_{(-)}^\pm$ are copies of the sets $\mathbb{R}_{(+)}$ and $\mathbb{R}_{(-)}$, respectively. Let us introduce a topology on $\Sigma$. 1. The fundamental system of neighbourhoods of the point $(\xi_0,\lambda_0) \in \mathbb{R}_d\times \mathbb{R}$ consists of the sets $\{\xi_0\} \times I$, where $I$ is an open interval containing $\lambda_0$. 2. For $\xi_0 \in \mathbb{R}^+_{(+)}$ distinct from $-\infty$ the fundamental system of neighbourhoods consists of the sets
$$
\begin{equation*}
\bigl(\{\xi_0\}\times (\lambda_1,+\infty) \cup (\xi_0,b)\times \mathbb{R}\bigr) \cup [\xi_0,b) \cup (\xi_0,b) \subset (\mathbb{R}_d\times \mathbb{R}) \cup \mathbb{R}_{(+)}^+ \cup \mathbb{R}_{(-)}^+,
\end{equation*}
\notag
$$
where $\lambda_1 \in \mathbb{R}$ and $b>\xi_0$. For $\xi_0=-\infty$ the set $\{\xi_0\}\times (\lambda_1,+\infty) \cup (\xi_0,b)\times \mathbb{R}$ is replaced by $(-\infty,b)\times \mathbb{R}$, where $b$ is arbitrary. 3. For $\xi_0 \in \mathbb{R}^+_{(-)}$ distinct from $+\infty$ the fundamental system of neighbourhoods consists of the sets
$$
\begin{equation*}
\bigl(\{\xi_0\}\times (-\infty,\lambda_1) \cup (a,\xi_0)\times \mathbb{R}\bigr) \cup (a,\xi_0) \cup (a,\xi_0] \subset (\mathbb{R}_d\times \mathbb{R}) \cup \mathbb{R}_{(+)}^+ \cup \mathbb{R}_{(-)}^+,
\end{equation*}
\notag
$$
where $\lambda_1 \in \mathbb{R}$ and $a<\xi_0$. For $\xi_0=+\infty$ the set $\{\xi_0\}\times (-\infty,\lambda_1) \cup (a,\xi_0)\times \mathbb{R}$ is replaced by $(a,+\infty)\times \mathbb{R}$, where $a$ is arbitrary. 4. For $\xi_0 \in \mathbb{R}^-_{(+)}$ distinct from $-\infty$ the fundamental system of neighbourhoods consists of the sets
$$
\begin{equation*}
\bigl(\{\xi_0\}\times (-\infty,\lambda_1) \cup (\xi_0,b)\times \mathbb{R}\bigr) \cup [\xi_0,b) \cup (\xi_0,b) \subset (\mathbb{R}_d\times \mathbb{R}) \cup \mathbb{R}_{(+)}^- \cup \mathbb{R}_{(-)}^-,
\end{equation*}
\notag
$$
where $\lambda_1 \in \mathbb{R}$ and $b>\xi_0$. For $\xi_0=-\infty$ the set $\{\xi_0\}\times (-\infty,\lambda_1) \cup (\xi_0,b)\times \mathbb{R}$ is replaced $(-\infty,b)\times \mathbb{R}$, where $b$ is arbitrary. 5. For $\xi_0 \in \mathbb{R}^-_{(-)}$ distinct from $+\infty$ the fundamental system of neighbourhoods consists of the sets
$$
\begin{equation*}
\bigl(\{\xi_0\}\times (\lambda_1,+\infty) \cup (a,\xi_0)\times \mathbb{R}\bigr) \cup (a,\xi_0) \cup (a,\xi_0] \subset (\mathbb{R}_d\times \mathbb{R}) \cup \mathbb{R}_{(+)}^- \cup \mathbb{R}_{(-)}^-,
\end{equation*}
\notag
$$
where $\lambda_1 \in \mathbb{R}$ and $a<\xi_0$. For $\xi_0=+\infty$ the set $\{\xi_0\}\times (\lambda_1,+\infty) \cup (a,\xi_0)\times \mathbb{R}$ is replaced by $(a,+\infty)\times \mathbb{R}$, where $a$ is arbitrary. We define a bijection $f\colon\Sigma \to \widehat{\mathcal{B}_\infty}$ by the formulae
$$
\begin{equation*}
\mathbb{R}_d \times \mathbb{R} \ni (\xi,\lambda) \mapsto [\pi(\xi,\lambda)]\quad\text{and} \quad \mathbb{R}^\pm_{(\omega)} \ni \xi \mapsto [\pi(\pm\infty,\omega,\xi)].
\end{equation*}
\notag
$$
Lemma 3. The topology on $\widehat{\mathcal{B}_\infty}$ induced by the bijection $f$ coincides with the Jacobson topology. § 5. The action of the group $ {\mathbb{R}}_d$ on the spectrum of $\mathcal{B}_\infty$Let $(\mathcal{A}, G, \alpha)$ be a dynamical system with discrete abelian group $G$. Clearly, if $\pi$ is an irreducible representation of the algebra $\mathcal{A}$, then $\pi\circ \alpha_t^{-1}$ is irreducible too. The induced action of an element $t$ of $G$ on $\widehat{\mathcal{A}}$ is defined by the formula
$$
\begin{equation}
[\pi]\mapsto [\pi\circ\alpha_t^{-1}], \qquad [\pi] \in \widehat{\mathcal{A}}.
\end{equation}
\tag{5.1}
$$
The action of $G$ on $\widehat{\mathcal{A}}$ obtained in this way is denoted by the same symbol $\alpha$. An action $\alpha$ of a group $G$ is called topologically free on $\mathcal{A}$ (see [18] and [19]) if for each finite subset $H\subset G\setminus\{e\}$ the set
$$
\begin{equation*}
\bigcap_{t\in H}\bigl\{[\pi] \in \widehat{\mathcal{A}}\colon \alpha_t([\pi])\neq [\pi]\bigr\}
\end{equation*}
\notag
$$
is dense in $\widehat{\mathcal{A}}$. Lemma 4. The action $\alpha$ of the group $\mathbb{R}_d$ is topologically free on ${\mathcal{B}_\infty}$. Proof. Let us describe the induced action of $\mathbb{R}_d$ for $(\mathcal{B}_\infty, \mathbb{R}_d,\alpha)$. Formulae (3.4), (3.5), (5.1) and Lemma 2 imply that $\mathbb{R}_d$ acts on the spectrum of $\mathcal{B}_\infty$ as follows:
$$
\begin{equation}
\alpha_k([\pi(\phi,\omega,\xi)])=[\pi(\phi,\omega,\xi-k)]
\end{equation}
\tag{5.3}
$$
for $\xi \neq \infty$ and
$$
\begin{equation}
\alpha_k ([\pi(\phi,\pm,\mp\infty)])=[\pi(\phi,\pm,\mp\infty)].
\end{equation}
\tag{5.4}
$$
Each of the representations involved in (5.2) and (5.3) satisfies the relation ${\alpha_k([\pi])\neq [\pi]}$ for each $k \in \mathbb{R}_d\setminus\{0\}$. Therefore, for any $H\subset \mathbb{R}_d\setminus\{0\}$ the following inclusion holds:
$$
\begin{equation*}
\begin{aligned} \, &\bigl\{[\pi(\xi,\lambda)]\colon \xi \in \mathbb{R}_d, \,\lambda \in \mathbb{R}\bigr\} \cup \bigl\{[\pi(\phi,\omega,\xi)]\colon \xi \neq \infty \bigr\} \\ &\qquad\subset \bigcap_{k\in H} \bigl\{[\pi] \in \widehat{\mathcal{B}_\infty}\colon \alpha_k([\pi])\neq [\pi]\bigr\}. \end{aligned}
\end{equation*}
\notag
$$
From Lemma 3 it follows that the set is dense in $\widehat{\mathcal{B}_\infty}$. Hence the set is dense in $\widehat{\mathcal{B}_\infty}$.
The lemma is proved. Let $(\mathcal{A},G,\alpha)$ be a dynamical system with discrete group $G$, and let $\iota$ be the natural embedding of $\mathcal{A}$ into $\mathcal{A}\rtimes_\alpha {G}$ defined by
$$
\begin{equation}
\iota\colon a \mapsto f(t)=\begin{cases} a, &t=e, \\ 0, & t \neq e. \end{cases}
\end{equation}
\tag{5.5}
$$
We say that an ideal $\mathcal{I} \subset \mathcal{A}\rtimes_{\alpha} G$ has a trivial intersection with $\mathcal{A}$ and write $\mathcal{I} \cap \mathcal{A}=\{0\}$ if $\mathcal{I} \cap \iota(\mathcal{A})=\{0\}$. Lemma 5. Let $(\mathcal{A},G,\alpha)$ be a dynamical system with discrete abelian group $G$ and topologically free action $\alpha$ on $\mathcal{A}$. If $\mathcal{I}$ is an ideal in $\mathcal{A}\rtimes_{\alpha} G$ such that $\mathcal{I}\cap \mathcal{A}=\{0\}$, then $\mathcal{I}=\{0\}$. Proof. It follows from [19], Theorem 1, that if the action $\alpha$ of the group $G$ is topologically free on $\mathcal{A}$, then any ideal $\mathcal{I} \subset \mathcal{A}\rtimes_\alpha G$ having a trivial intersection with $\mathcal{A}$ is contained in the kernel of the natural surjection
$$
\begin{equation}
\mathcal{A} \rtimes_{\alpha} G \to \mathcal{A} \rtimes_{r,\alpha} G.
\end{equation}
\tag{5.6}
$$
Since the group $G$ is abelian, it is amenable, and thus the reduced crossed product coincides with the full one: $\mathcal{A}\rtimes_{\alpha,r} G \cong \mathcal{A}\rtimes_{\alpha} G$ (see [17], Theorem 7.13). Therefore, the kernel of mapping (5.6) is trivial, and $\mathcal{I}=\{0\}$. Completion of the proof of Theorem 1. We set $\mathcal{I}=\ker\Phi_U$ and verify the conditions of Lemma 5. The triviality of the intersection of $\mathcal{I}$ with $\mathcal{B}_\infty$ follows from the equality $\Phi_U \circ \iota=\operatorname{id}_{\mathcal{B}_\infty}$, where $\operatorname{id}_{\mathcal{B}_\infty}$ is the identity map on $\mathcal{B}_\infty$. According to Lemma 4, the action $\alpha$ is topologically free on $\mathcal{B}_\infty$. Thus, Lemma 5 implies that $\ker \Phi_U={0}$. The theorem is proved. Remark 2. The triple $(\mathcal{B},\mathbb{R}_d,\alpha')$, where is also a dynamical system. However, $\mathcal{B}\rtimes_{\alpha'}\mathbb{R}_d$ is not isomorphic to $\mathfrak{T}$, since the mapping analogous to $\Phi_U$ in the proof of Theorem 1 has a nontrivial kernel. § 6. The space of quasi-orbits of $(\mathcal{B}_\infty, {\mathbb{R}}_d,\alpha)$The notion of ‘quasi-orbits’ in the primitive spectrum of the algebra $\mathcal{A}$ is needed in the next section, where we describe the connection between irreducible representations of $\mathcal{A}\rtimes_{\alpha} G$ and certain ideals of $\mathcal{A}$ invariant under the action $\alpha$. Let $(\mathcal{A},G,\alpha)$ be a dynamical system, and let $P$ be a primitive ideal of $\mathcal{A}$. The set $\{\alpha_t(P)\colon t \in G\}$ is called the orbit of $P$ in $\operatorname{Prim} \mathcal{A}$. We consider two orbits to be equivalent if their closures in $\operatorname{Prim} \mathcal{A}$ coincide. The corresponding equivalence classes are called quasi-orbits. Let us provide another, more convenient definition of quasi-orbits by interpreting them not as classes of orbits, but as classes of ideals in $\operatorname{Prim}\mathcal{A}$. We say that $P_1, P_2 \in \operatorname{Prim}\mathcal{A}$ belong to the same quasi-orbit, denoted by $P_1 \sim_q P_2$, if
$$
\begin{equation*}
\overline{\{\alpha_t(P_1)\colon t \in G\}}=\overline{\{\alpha_t(P_2)\colon t \in G\}}.
\end{equation*}
\notag
$$
It is obvious that $\sim_q$ is an equivalence relation on $\operatorname{Prim}\mathcal{A}$. The quotient space $\mathcal{Q}=\operatorname{Prim}\mathcal{A}/{\sim_q}$ is called the space of quasi-orbits of the dynamical system $(\mathcal{A}, G, \alpha)$. It must be emphasized that if $\kappa\colon \operatorname{Prim}\mathcal{A} \to \operatorname{Prim}\mathcal{A}/{\sim_q}$ is the corresponding quotient map, then for any quasi-orbit $Q$ its preimage $\kappa^{-1}(Q)$ is a union of orbits. Moreover, if $P \in \kappa^{-1}(Q)$, then the set $\overline{\{\alpha_t(P)\colon t \in G\}}$ contains $\kappa^{-1}(Q)$, but generally does not coincide with it. Let us describe the space of quasi-orbits of $(\mathcal{B}_\infty, \mathbb{R}_d,\alpha)$. From the results of § 4 it follows that any irreducible representation of $\mathcal{B}_\infty$ is finite-dimensional, and therefore $\mathcal{B}_\infty$ is a GCR-algebra. Consequently (see [20], § 4.4.1), $\operatorname{Prim}\mathcal{B}_\infty$ and $\widehat{\mathcal{B}_\infty}$ are connected by the homeomorphism
$$
\begin{equation*}
\widehat{\mathcal{B}_\infty} \to \operatorname{Prim}\mathcal{B}_\infty\colon [\pi] \mapsto \ker\pi.
\end{equation*}
\notag
$$
This homeomorphism commutes with $\alpha$, since $\ker (\pi\circ \alpha_t^{-1})=\alpha_t(\ker\pi) $ (see (5.1)). In turn, by Lemma 3 the elements of $\widehat{\mathcal{B}_\infty}$ are parameterized by points of $\Sigma$ in (4.9). The action on $\Sigma$ is induced by the action on $\widehat{\mathcal{B}_\infty}$ (see (5.2)–(5.4)) by means of the homeomorphism $f$ from Lemma 3. We say that $\sigma_{1,2} \in \Sigma$ belong to the same quasi-orbit if $\overline{\{\alpha_t(\sigma_1)\colon t \in G\}}= \overline{\{\alpha_t(\sigma_2)\colon t \in G\}}$; we denote the space of quasi-orbits on $\Sigma$ by $\mathcal{Q}_\Sigma$. Then the space $\mathcal{Q}$ of quasi-orbits on $\operatorname{Prim}\mathcal{B}_\infty$ is homeomorphic to $\mathcal{Q}_\Sigma$. First let us describe the orbits on $\Sigma$ and their closures. In view of (5.2) the orbit of $(\xi,\lambda) \in \mathbb{R}_d\times \mathbb{R}\subset \Sigma$ is the set $\mathbb{R}_d \times \{\lambda\}$. It follows from the description of the topology on $\Sigma$ before Lemma 3 that the closure of this set coincides with
$$
\begin{equation*}
(\mathbb{R}_d\times \{\lambda\}) \cup\mathbb{R}_{(+)}^+\cup \mathbb{R}_{(-)}^+\cup \mathbb{R}_{(+)}^- \cup \mathbb{R}_{(-)}^- \subset \Sigma.
\end{equation*}
\notag
$$
Indeed, only points of the form $(\xi_0,\lambda_0)$ with $\lambda_0\neq \lambda$ have neighbourhoods disjoint from $\mathbb{R}_d \times \{\lambda\}$. Further, by equality (5.3) the orbit of $\xi \in \mathbb{R}^{+}_{(\omega)}$ for $\xi\neq \infty$ is $\{\xi \in \mathbb{R}^{+}_{(\omega)}\colon \xi \neq \infty\}$, and its closure coincides with the set $\mathbb{R}^{+}_{(+)} \cup \mathbb{R}^{+}_{(-)}$. Similarly, the orbit of a point $\xi \in \mathbb{R}^{-}_{(\omega)}$ for $\xi\neq \infty$ is $\{\xi \in \mathbb{R}^{-}_{(\omega)}\colon \xi \neq \infty\}$, and its closure coincides with the set $\mathbb{R}^{-}_{(+)} \cup \mathbb{R}^{-}_{(-)}$. Finally, by (5.4) the points $-\infty \in \mathbb{R}^{\pm}_{(+)}$ and $+\infty \in \mathbb{R}^{\pm}_{(-)}$ are fixed, that is, their orbits are $\{-\infty\} \subset \mathbb{R}^{\pm}_{(+)}$ and $\{+\infty\}\subset \mathbb{R}^{\pm}_{(-)}$, respectively, and these orbits are closed. Thus, points $(\xi,\lambda) \in \Sigma$ with fixed $\lambda$ and various $\xi\in \mathbb{R}_d$ belong to the same quasi-orbit, which we denote by $Q_1(\lambda)$. Points $\xi \in \mathbb{R}^+_{(\omega)}$ with $\xi \neq \infty $ and $\omega=\pm$ belong to the same quasi-orbit, denoted by $Q_2(+\infty)$. Similarly, points $\xi \in \mathbb{R}^-_{(\omega)}$ with $\xi \neq \infty $ and $\omega=\pm$ belong to the same quasi-orbit $Q_2(-\infty)$. Finally, quasi-orbits associated with $-\infty \in \mathbb{R}^{\pm}_{(+)}$ and $+\infty \in \mathbb{R}^{\pm}_{(-)}$ are denoted by $Q_3(\pm \infty,-\infty)$ and $Q_3(\pm \infty,+\infty)$, respectively. In summary, we have
$$
\begin{equation*}
\mathcal{Q}_\Sigma=\bigl\{Q_1(\lambda)\colon \lambda \in \mathbb{R}\bigr\} \cup \bigl\{Q_2(\phi)\colon \phi= \pm \infty\bigr\} \cup \bigl\{Q_3(\phi,\psi)\colon \phi,\psi=\pm \infty\bigr\}.
\end{equation*}
\notag
$$
Let $\kappa_\Sigma\colon \Sigma \to \mathcal{Q}_\Sigma$ be the quotient map. In what follows we need a detailed description of the sets $\kappa^{-1}_\Sigma(Q)$ and $\overline{\kappa_{\Sigma}^{-1}(Q)}$ for each $Q \in \mathcal{Q}_\Sigma$. The following relations hold:
$$
\begin{equation}
\kappa_{\Sigma}^{-1}(Q_1(\lambda_0))=\mathbb{R}_d\times \{\lambda_0\},
\end{equation}
\tag{6.1}
$$
$$
\begin{equation}
\overline{\kappa_{\Sigma}^{-1}(Q_1(\lambda_0))}= (\mathbb{R}_d\times \{\lambda_0\}) \cup\mathbb{R}_{(+)}^+\cup \mathbb{R}_{(-)}^+\cup \mathbb{R}_{(+)}^- \cup \mathbb{R}_{(-)}^-,
\end{equation}
\tag{6.2}
$$
$$
\begin{equation}
\kappa_{\Sigma}^{-1}(Q_2(\pm \infty))=\bigl\{\xi \in \mathbb{R}^{\pm}_{(+)}\cup \mathbb{R}^{\pm}_{(-)} \colon \xi \neq \infty\bigr\},
\end{equation}
\tag{6.3}
$$
$$
\begin{equation}
\overline{ \kappa_{\Sigma}^{-1}(Q_2(\pm\infty)) }=\mathbb{R}^{\pm}_{(+)} \cup \mathbb{R}^{\pm}_{(-)},
\end{equation}
\tag{6.4}
$$
$$
\begin{equation}
\kappa_{\Sigma}^{-1}(Q_3(\pm \infty,-\infty))=\overline{ \kappa_{\Sigma}^{-1}(Q_3(\pm \infty,-\infty)) }=\{-\infty\} \subset \mathbb{R}^{\pm}_{(+)},
\end{equation}
\tag{6.5}
$$
$$
\begin{equation}
\kappa_{\Sigma}^{-1}(Q_3(\pm \infty,+\infty))=\overline{ \kappa_{\Sigma}^{-1}(Q_3(\pm \infty,+\infty)) }=\{+\infty\} \subset \mathbb{R}^{\pm}_{(-)}.
\end{equation}
\tag{6.6}
$$
Now we describe the topology on $\mathcal{Q}_\Sigma$. According to [17], Lemma 6.12, the quotient map $\kappa\colon \operatorname{Prim} \mathcal{B}_\infty \to \mathcal{Q}$ is open, hence the map $\kappa_\Sigma\colon \Sigma \to \mathcal{Q}_\Sigma$ is also open. Therefore, the family of sets $\{\kappa_\Sigma(U)\}$, where $U$ ranges over the neighbourhoods introduced in paragraphs 1–5 before Lemma 3, forms a fundamental system of neighbourhoods in the space $\mathcal{Q}_\Sigma$. Thus, the fundamental system of neighbourhoods of the quasi-orbit $Q_1(\lambda_0)$ consists of the sets where $a,b \in \mathbb{R}$ and $a<\lambda_0<b$; the fundamental system of neighbourhoods of the quasi-orbit $Q_2(\phi)$ consists of the sets
$$
\begin{equation}
\{Q_1(\lambda)\colon \lambda \in \mathbb{R}\} \cup \{ Q_2(\phi)\};
\end{equation}
\tag{6.8}
$$
and the fundamental system of neighbourhoods of the quasi-orbit $Q_3(\phi, \pm\infty)$ consists of the sets
$$
\begin{equation}
\{Q_1(\lambda)\colon \lambda \in \mathbb{R}\}\cup \{ Q_2(\phi)\} \cup \{Q_3(\phi,\pm\infty)\}.
\end{equation}
\tag{6.9}
$$
For any algebra $\mathcal{A}$ the space $\operatorname{Prim}\mathcal{A}$ is locally compact (see [20], § 3.3.8). Since $\kappa\colon \operatorname{Prim}\mathcal{A} \to \mathcal{Q}$ is open and continuous, the space of quasi-orbits $\mathcal{Q}$ is locally compact. A locally compact topological space is called almost Hausdorff if every locally compact subspace contains a nonempty relatively open Hausdorff subset. It is obvious that a locally compact subspace of an almost Hausdorff space is almost Hausdorff. Lemma 6. The space of quasi-orbits $\mathcal{Q}=\operatorname{Prim}\mathcal{B}_\infty/{\sim_q}$ is almost Hausdorff. Proof. It suffices to present a sequence of increasing open sets $U_n$ in $\mathcal{Q}_\Sigma$ such that $U_0=\varnothing$, $U_N=\mathcal{Q}_\Sigma$, and for every $n=0,1,\dots, N-1$ the sets $U_{n+1} \setminus U_n$ are dense Hausdorff subspaces of $\mathcal{Q}_\Sigma \setminus U_n$ (see [17], Lemma 6.3, (d)). Let
$$
\begin{equation*}
\begin{gathered} \, U_1=\{Q_1(\lambda)\colon \lambda \in \mathbb{R}\}, \\ U_2=\{Q_1(\lambda)\colon \lambda \in \mathbb{R}\} \cup \{ Q_2(\phi)\colon \phi=\pm \infty \} \end{gathered}
\end{equation*}
\notag
$$
and The point $Q_1(\lambda) $ belongs to $U_1$ and $U_2$ together with a neighbourhood of the form (6.7); the point $Q_2(\phi)$ belongs to the set $U_2$ together with a neighbourhood of the form (6.8). Hence the sets $U_1$ and $U_2$ are open. The set $U_1$ with the induced topology is homeomorphic to $\mathbb{R}$; thus, it is Hausdorff. In addition, the set $U_1$ is dense in $Q_\Sigma$, since every neighbourhood of a point in the complement of $U_1$ contains a neighbourhood of the form (6.8) or (6.9) and thus intersects $U_1$. The set $U_2\setminus U_1$ with the induced topology is discrete, hence Hausdorff. Every neighbourhood of a point in $\mathcal{Q}_\Sigma\setminus U_1$ contains a neighbourhood of the form (6.8) or (6.9), and thus intersects $U_2\setminus U_1$, that is, $U_2\setminus U_1$ is dense in $\mathcal{Q}_\Sigma\setminus U_1$. Finally, the set $U_3 \setminus U_2=\mathcal{Q}_\Sigma \setminus U_2$ with the induced topology is discrete and therefore Hausdorff. The question of the density of $U_3 \setminus U_2$ in $\mathcal{Q}_\Sigma \setminus U_2$ is trivial, since these sets coincide.
The lemma is proved. § 7. Localization principle on quasi-orbitsLet $(\pi,U)$ be a covariant representation of a dynamical system $(\mathcal{A},G,\alpha)$. Then the representation $\pi\rtimes U$ of the $*$-algebra $C_c(G,\mathcal{A})$ defined by (3.1) extends by continuity to a representation of $\mathcal{A}\rtimes_{\alpha}G$. We denote this extension by the same symbol $\pi\rtimes U$. If $(\pi, U)$ is a nondegenerate covariant representation (that is, $\pi$ is nondegenerate), then $\pi \rtimes U$ is a nondegenerate representation of $\mathcal{A}\rtimes_{\alpha}G$ (see [17], Proposition 2.40). Conversely, for any nondegenerate representation $L$ of $\mathcal{A}\rtimes_{\alpha}G$ there exists a unique nondegenerate covariant representation $(\pi,U)$ of $(\mathcal{A},G,\alpha)$ such that $L$ coincides with $\pi \rtimes U$ (see [17], Proposition 2.40). Therefore, in what follows we consider only representations of the form $\pi \rtimes U$. Let $\mathcal{I}$ be an ideal of $\mathcal{A}$ invariant under $\alpha$ and $\rho\colon \mathcal{A} \to \mathcal{A}/\mathcal{I}$ be the quotient map. Let $(\pi, U)$ be a nondegenerate covariant representation of $(\mathcal{A},G,\alpha)$ satisfying $\mathcal{I} \,{\subset} \ker \pi $. As usual, we define the quotient representation $\pi'$ of $\mathcal{A}/\mathcal{I}$ by $\pi=\pi' \circ \rho$. The action $\alpha^{\mathcal{I}}$ of $G$ on the quotient $\mathcal{A}/\mathcal{I}$ is well defined by $\alpha_t^\mathcal{I}(\rho(a))=\rho(\alpha_t(a))$. Then the pair $(\pi', U)$ forms a nondegenerate covariant representation of $(\mathcal{A}/\mathcal{I},G,\alpha^\mathcal{I})$. Thus, for every nondegenerate representation $L=\pi \rtimes U$ of $\mathcal{A}\rtimes_\alpha G$ such that $\mathcal{I} \subset \ker \pi$ there is a naturally corresponding nondegenerate representation $L'=\pi'\rtimes U$ of $(\mathcal{A}/ \mathcal{I}) \rtimes_{\alpha^\mathcal{I}} G$. Conversely, if $L'=\pi'\rtimes U$ is a nondegenerate representation of $(\mathcal{A}/\mathcal{I})\rtimes_{\alpha^\mathcal{I}} G$, then $\pi=\pi'\circ \rho$ is a nondegenerate representation of $\mathcal{A}$, and $\pi \rtimes U$ is a nondegenerate representation of $\mathcal{A}\rtimes_{\alpha} G$. Recall that, by definition, a covariant representation $(\pi,U)$ of $(\mathcal{A},G,\alpha)$ is irreducible if every closed subspace $\mathcal{F}\subset \mathcal{H}$ satisfying $\pi(a)\mathcal{F}\subset \mathcal{F}$ and $U(t)\mathcal{F}\subset \mathcal{F}$ for all $a\in\mathcal{A}$ and $t\in G$ coincides with $\{0\}$ or $\mathcal{H}$. It is clear that the covariant representations $(\pi'\circ \rho,U)$ and $(\pi',U)$ of $(\mathcal{A},G,\alpha)$ and $(\mathcal{A}/\mathcal{I},G,\alpha^\mathcal{I})$, respectively, are reducible or irreducible simultaneously. According to [17], Proposition 2.40, a representation $\pi\rtimes U$ of $\mathcal{A}\rtimes_\alpha G$ is irreducible if and only if $(\pi,U)$ is. From all the above we conclude that the spectrum of $(\mathcal{A}/\mathcal{I})\rtimes_{\alpha^\mathcal{I}}G$ is naturally identified with a part of the spectrum $(\mathcal{A}\rtimes_{\alpha} G ) \,\widehat{}$ . Let $\mathcal{Q}$ be the space of quasi-orbits of $(\mathcal{A}, G, \alpha)$, and let $\kappa\colon \operatorname{Prim} \mathcal{A} \to \mathcal{Q}$ be the corresponding quotient map. For each quasi-orbit $Q\in \mathcal{Q}$ we introduce the ideal $\mathcal{I}_Q=\bigcap_{P \in \kappa^{-1}(Q)} P $ and the quotient $\mathcal{A}_Q=\mathcal{A}/\mathcal{I}_Q$. Note that $\mathcal{I}_Q$ is an invariant ideal of $\mathcal{A}$, and consider the action $\alpha^Q=\alpha^{\mathcal{I}_Q}$. Identifying the spectrum $(\mathcal{A}_Q \rtimes_{\alpha^Q}G)\,\widehat{}$ with a subset of $(\mathcal{A} \rtimes_{\alpha} G)\,\widehat{}$ we have
$$
\begin{equation*}
(\mathcal{A} \rtimes_{\alpha} G)\,\widehat{}\ \supset \bigcup_{Q \in \mathcal{Q}} (\mathcal{A}_Q \rtimes_{\alpha^Q}G)\,\widehat{}.
\end{equation*}
\notag
$$
Lemma 7. Suppose that the space of quasi-orbits $\mathcal{Q}$ of a dynamical system $(\mathcal{A}, G, \alpha)$ is almost Hausdorff. Then Proof. Since $\mathcal{Q}$ is almost Hausdorff, for any irreducible representation $L=\pi \rtimes U$ of $\mathcal{A}\rtimes_\alpha G$ there exists $Q_\pi \in \mathcal{Q}$ such that see [17], Proposition 6.21 and Definition 6.17. Hence $[L] \in (\mathcal{A}_{Q_\pi} \rtimes _{\alpha^{Q_\pi}}G)\,\widehat{}$ .
The lemma is proved. Suppose that the assumptions of Lemma 7 are fulfilled. Let $L=\pi\rtimes U$ be an irreducible representation of $\mathcal{A}\rtimes_\alpha G$, and let $L'=\pi'\rtimes U$ be the representation of $\mathcal{A}_{Q_\pi} \rtimes_{\alpha^{Q_\pi}}G$ corresponding to $L$. Equality (7.2) means that $\ker \pi'=\{0\}$, that is, $\pi'$ is faithful. Thus, within each set $(\mathcal{A}_Q \rtimes_{\alpha^Q}G)\,\widehat{}$ in (7.1) it suffices to select only those representations that are faithful on $\mathcal{A}_Q$. It turns out that all such representations can be described as representations of some ideal in $\mathcal{A}_Q \rtimes_{\alpha^Q}G$. To do this we need the following lemma. Lemma 8. Let $(\mathcal{A}, G, \alpha)$ be a dynamical system, $\mathcal{Q}$ be its space of quasi-orbits, and $\kappa \colon \operatorname{Prim} \mathcal{A} \to \mathcal{Q}$ be the quotient map. If $\mathcal{Q}$ is almost Hausdorff, then for any $Q \in \mathcal{Q}$ the set $\kappa^{-1}(\{Q\})$ is relatively open in its closure $\overline{\kappa^{-1}(\{Q\})}$. Proof. Repeating the proof of the implication (e) $\Rightarrow$ (b) in the proof of Theorem 6.2 in [17] one can show that, because of the almost Hausdorff property of $\mathcal{Q}$, the set $\{Q\}$ is relatively open in $\overline{\{Q\}}$. Since the map $\kappa$ is open, it follows that $\kappa^{-1}(\overline{\{Q\}})=\overline{\kappa^{-1}(Q)}$. Therefore, $\kappa^{-1}(Q)$ is relatively open in its closure.
The lemma is proved. For each quasi-orbit $Q \in \mathcal{Q}$ we consider the ideal $\mathcal{J}_Q=\bigcap_{P \in \overline{\kappa^{-1}(Q)} \setminus \kappa^{-1}(Q)} P$ of $\mathcal{A}$. If $\overline{\kappa^{-1}(Q)} \setminus \kappa^{-1}(Q)=\varnothing$, then we set $\mathcal{J}_Q=\mathcal{A}$. Recall that by the definition of the Jacobson topology the ideals $\bigcap_{P \in A} P$ and $\bigcap_{P \in B} P$ (for $A, B \subset \operatorname{Prim} \mathcal{A}$) coincide if and only if $\overline{A}=\overline{B}$. In particular, $\mathcal{I}_Q=\bigcap_{P \in \kappa^{-1}(Q)} P=\bigcap_{P \in \overline{\kappa^{-1}(Q)}} P$, and it is obvious that $\mathcal{I}_Q \subset \mathcal{J}_Q$. Lemma 8 implies that $\overline{\kappa^{-1}(Q)} \setminus \kappa^{-1}(Q)$ is closed in $\operatorname{Prim} \mathcal{A}$. Hence $\mathcal{I}_Q \neq \mathcal{J}_Q$, and the quotient $\widetilde{\mathcal{J}}_Q=\mathcal{J}_Q / \mathcal{I}_Q$ is nontrivial. Let $\rho_Q \colon \mathcal{A} \to \mathcal{A}_Q=\mathcal{A}/\mathcal{I}_Q$ be the quotient map; the algebra $\widetilde{\mathcal{J}}_Q$ is naturally isomorphic to the ideal $\rho_Q(\mathcal{J}_Q)$ of $\mathcal{A}_Q$. In what follows we identify $\widetilde{\mathcal{J}}_Q$ with $\rho_Q(\mathcal{J}_Q)$. The ideal $\mathcal{J}_Q$ is invariant under the action $\alpha$ on $\mathcal{A}$; hence, $\widetilde{\mathcal{J}}_Q$ is invariant under $\alpha^Q$, and the triple $(\widetilde{\mathcal{J}}_Q,G,\alpha^Q)$ is a dynamical system. According to [17], Lemma 3.17, there exists a natural embedding of $\widetilde{\mathcal{J}}_Q \rtimes_{\alpha^Q} G$ into $\mathcal{A}_Q \rtimes_{\alpha^Q} G$ as an ideal. We denote the image of this embedding by $\widetilde{\mathcal{J}}_Q \rtimes_{\alpha^Q} G$ again. The spectrum $(\widetilde{\mathcal{J}}_Q \rtimes_{\alpha^Q}G)\,\widehat{}$ is naturally identified with a part of the spectrum $(\mathcal{A}_Q \rtimes_{\alpha^Q}G)\,\widehat{}$ , which, in turn, is considered as a part of $(\mathcal{A} \rtimes_{\alpha} G)\,\widehat{}$ . Theorem 2. Suppose that the space of quasi-orbits $\mathcal{Q}$ of a dynamical system $(\mathcal{A}, G, \alpha)$ is almost Hausdorff. Then where the union is disjoint. Proof. Since $\widetilde{\mathcal{J}}_Q\rtimes_{\alpha^Q}G$ is an ideal of $\mathcal{A}_Q\rtimes_{\alpha^Q}G$, we have
$$
\begin{equation}
(\mathcal{A}_Q\rtimes_{\alpha^Q}G)\,\widehat{}= \biggl(\frac{\mathcal{A}_Q\rtimes_{\alpha^Q}G}{\widetilde{\mathcal{J}}_Q \rtimes_{\alpha^Q}G}\biggr) \,\widehat{} \ \cup(\widetilde{\mathcal{J}}_Q \rtimes_{\alpha^Q}G)\,\widehat{} \,.
\end{equation}
\tag{7.4}
$$
Any representation in the first set on the right-hand side of (7.4) is not faithful on $\mathcal{A}_Q$. Indeed, suppose that $L'=\pi'\rtimes U$ is such a representation; then its kernel contains the ideal $\widetilde{\mathcal{J}}_Q\rtimes_{\alpha^Q}G$, implying that $\widetilde{\mathcal{J}}_Q \subset \ker \pi'$. Hence (7.3) holds by Lemma 7 and the reasoning following it.
Let $L=\pi\rtimes U$ be a representation of $\mathcal{A} \rtimes_{\alpha} G$ from the set $(\widetilde{\mathcal{J}}_Q \rtimes_{\alpha^Q}G)\,\widehat{}$ (see (7.3)). We show that $\ker \pi=\mathcal{I}_Q$. Let $K\subset \operatorname{Prim} \mathcal{A}$ be the set of all primitive ideals containing $\ker \pi$. Since $\pi$ is a nontrivial representation, $K$ is nonempty. Moreover, $K$ is a closed set invariant under $\alpha$ and $\ker \pi=\bigcap_{P\in K} P$. Since $\pi$ corresponds to some representation of the algebra $\widetilde{\mathcal{J}}_Q=\mathcal{J}_Q/\mathcal{I}_Q$, we have $\mathcal{I}_Q \subset \ker \pi$. Thus, $K\subset\overline{\kappa^{-1}(Q)} $. Suppose $\ker \pi \neq \mathcal{I}_Q$, so that $K \neq \overline{\kappa^{-1}(Q)} $. The sets $\kappa^{-1}(Q)$ and $K$ are disjoint; otherwise, by the invariance of $K$, $\overline{\kappa^{-1}(Q)}\subset K$, implying that $\overline{\kappa^{-1}(Q)}=K$. Therefore, $K \subset\overline{\kappa^{-1}(Q)} \setminus \kappa^{-1}(Q)$, that is, $\ker \pi \supset \mathcal{J}_Q $. This means that the corresponding representation of $\widetilde{\mathcal{J}}_Q$ is trivial. The contradiction obtained proves that $\ker \pi=\mathcal{I}_Q$. If a representation $L=\pi\rtimes U$ of $\mathcal{A}\rtimes_{\alpha} G$ belongs simultaneously to $(\widetilde{\mathcal{J}}_{Q_1} \rtimes_{\alpha^{Q_1}}G)\,\widehat{}$ and $(\widetilde{\mathcal{J}}_{Q_2} \rtimes_{\alpha^{Q_2}}G)\,\widehat{}$ , then $\ker \pi=\mathcal{I}_{Q_1}=\mathcal{I}_{Q_2}$. By the definition of the space of quasi-orbits, the ideals $\mathcal{I}_{Q_1}$ and $\mathcal{I}_{Q_2}$ coincide if and only if $Q_1=Q_2$. Therefore, the union on the right-hand side of (7.3) is disjoint. The theorem is proved. § 8. The primitive spectrum of the local algebra $\mathfrak{T}_\infty$Now we proceed to the description of the spectrum of $\mathfrak{T}_\infty=\mathcal{B}_\infty \rtimes_\alpha \mathbb{R}_d$. According to Lemma 6, the space $\mathcal{Q}=\operatorname{Prim}\mathcal{B}_\infty/{\sim_q}$ is almost Hausdorff, and thus, by Theorem 2, the following relation holds:
$$
\begin{equation*}
(\mathcal{B}_\infty \rtimes_{\alpha} \mathbb{R}_d)\,\widehat{}\ =\bigcup_{Q \in \mathcal{Q}} (\widetilde{\mathcal{J}}_Q \rtimes_{\alpha^Q}\mathbb{R}_d)\,\widehat{}\ ,
\end{equation*}
\notag
$$
where $\widetilde{\mathcal{J}}_Q=\mathcal{J}_Q/\mathcal{I}_Q$. To describe the ideals $\mathcal{J}_Q$ and $\mathcal{I}_Q$ it is convenient to introduce the symbol $\Phi_\infty$ of an element $[b]_\infty \in \mathcal{B}_\infty$. We define the function $\Phi_\infty([b]_\infty)$ on $\Sigma$ by
$$
\begin{equation*}
\begin{gathered} \, \mathbb{R}_d \times \mathbb{R} \ni (\xi,\lambda) \mapsto \pi(\xi,\lambda)([b]_\infty), \\ \mathbb{R}^\pm_{(\omega)} \ni \xi \mapsto \pi(\pm\infty,\omega,\xi)([b]_\infty). \end{gathered}
\end{equation*}
\notag
$$
The action $\alpha$ of $\mathbb{R}_d$ on $\Sigma$ induces an action $\alpha$ of $\mathbb{R}_d$ on symbols by the rule $\alpha_k(\Phi_\infty([b]_\infty))=\Phi_\infty ([b]_\infty) \circ \alpha_{-k}$, $k \in \mathbb{R}_d$. Then the action on symbols is consistent with the action on elements of $\mathcal{B}_\infty$, that is, the following equality holds:
$$
\begin{equation*}
(\Phi_\infty \circ \alpha_k)([b]_\infty)=(\alpha_k \circ \Phi_\infty)([b]_\infty).
\end{equation*}
\notag
$$
Let $\mathcal{J}_{\mathrm{com}}$ be the ideal generated by the commutators in $\mathcal{B}_\infty$. It is clear that the symbols of elements of $\mathcal{J}_{\mathrm{com}}$ vanish at points in $\Sigma$ corresponding to one-dimensional representations. From [8], Theorem 4.15, it follows that the map $[b]_\infty \mapsto \Phi_\infty([b]_\infty)$ induces isomorphisms
$$
\begin{equation}
\mathcal{J}_{\mathrm{com}} \cong C_0(\mathbb{R}_d\times \mathbb{R})\otimes M_2(\mathbb{C})
\end{equation}
\tag{8.1}
$$
and
$$
\begin{equation}
\mathcal{B}_\infty /\mathcal{J}_{\mathrm{com}} \cong C(\mathbb{R}_{(+)}^+\cup \mathbb{R}_{(-)}^+\cup \mathbb{R}_{(+)}^- \cup \mathbb{R}_{(-)}^-),
\end{equation}
\tag{8.2}
$$
where $M_2(\mathbb{C})$ is the algebra of $2\times 2$ matrices, and the topology on $\mathbb{R}_{(+)}^+\cup \mathbb{R}_{(-)}^+\cup \mathbb{R}_{(+)}^- \cup \mathbb{R}_{(-)}^-$ is induced from $\Sigma$. Let us describe the algebras $\widetilde{\mathcal{J}}_Q \rtimes_{\alpha^Q}\mathbb{R}_d$ for $Q=Q_1(\lambda_0)$ (to simplify the notation we identify the quasi-orbit spaces $\mathcal{Q}$ and $\mathcal{Q}_\Sigma$). From (6.1)-(6.2) and (8.1) it follows that $\mathcal{J}_{Q_1(\lambda_0)}=\mathcal{J}_{\mathrm{com}}\cong C_0(\mathbb{R}_d \times \mathbb{R})\otimes M_2(\mathbb{C})$, and the ideal $\mathcal{I}_{Q_1(\lambda_0)}$ consists of the elements of $\mathcal{J}_{Q_1(\lambda_0)}$ whose symbols vanish on the set $\mathbb{R}_d\times \{\lambda_0\}$. Therefore, $\Phi_\infty$ induces an isomorphism
$$
\begin{equation}
[b]_\infty+\mathcal{I}_{Q_1(\lambda_0)} \mapsto \pi(\,\cdot\,,\lambda_0)([b]_\infty )\colon\widetilde{\mathcal{J}}_{Q_1(\lambda_0)}=\mathcal{J}_{Q_1(\lambda_0)} / \mathcal{I}_{Q_1(\lambda_0)} \cong C_0(\mathbb{R}_d )\otimes M_2(\mathbb{C}).
\end{equation}
\tag{8.3}
$$
It follows from (5.2) that the action $\alpha$ of the group $\mathbb{R}_d$ results in a shift of the argument of any function in $C_0(\mathbb{R}_d)\otimes M_2(\mathbb{C})$ and leaves its values unchanged. Therefore, by [17], Lemma 2.75, there is an isomorphism
$$
\begin{equation*}
\bigl( C_0(\mathbb{R}_d)\otimes M_2(\mathbb{C}) \bigr) \rtimes_{\alpha} \mathbb{R}_d \cong (C_0(\mathbb{R}_d) \rtimes_{\alpha} \mathbb{R}_d )\otimes M_2(\mathbb{C}),
\end{equation*}
\notag
$$
and by [21], Theorem II.10.4.3, there is an isomorphism $C_0(\mathbb{R}_d) \rtimes_{\alpha}\mathbb{R}_d \cong \mathrm{KL}_2(\mathbb{R}_d)$. Thus, we have
$$
\begin{equation}
\widetilde{\mathcal{J}}_{Q_1(\lambda_0)} \rtimes_{\alpha^{Q_1(\lambda_0)}}\mathbb{R}_d \cong \mathrm{KL}_2(\mathbb{R}_d)\otimes M_2(\mathbb{C}),
\end{equation}
\tag{8.4}
$$
and therefore the spectrum $(\widetilde{\mathcal{J}}_{Q_1(\lambda_0)} \rtimes_{\alpha^{Q_1(\lambda_0)}}\mathbb{R}_d)\,\widehat{}$ consists of a single element. Let $Q=Q_2(+\infty)$. From (6.3), (6.4) it follows that $\mathcal{I}_{Q_2(+\infty)}$ consists of elements $[b]_\infty$ whose symbols vanish on $\mathbb{R}_{(+)}^+ \cup \mathbb{R}_{(-)}^+$. Therefore, $\mathcal{J}_{\mathrm{com}}\subset \mathcal{I}_{Q_2(+\infty)} $, and by (8.2) the quotient algebra $\mathcal{A}_{Q_2(+\infty)}= \mathcal{B}_\infty/\mathcal{I}_{Q_2(+\infty)}$ is isomorphic to $C(\mathbb{R}_{(+)}^+ \cup \mathbb{R}_{(-)}^+)$. According to (6.3), (6.4), the ideal $\widetilde{\mathcal{J}}_{Q_2(+\infty)}$ is isomorphic to $C_0\bigl((\mathbb{R}_{(+)}^{+} \setminus\{-\infty\})\cup (\mathbb{R}_{(-)}^{+}\setminus \{+\infty\})\bigr)$. The action $\alpha$ of the group $\mathbb{R}_d$ on $C_0\bigl((\mathbb{R}_{(+)}^{+} \setminus\{-\infty\})\cup (\mathbb{R}_{(-)}^{+}\setminus \{+\infty\})\bigr)$ is defined by (5.3). We have
$$
\begin{equation}
\widetilde{\mathcal{J}}_{Q_2(+\infty)} \rtimes_{\alpha^{Q_2(+\infty)}}\mathbb{R}_d \cong C_0\bigl((\mathbb{R}_{(+)}^{+} \setminus\{-\infty\})\cup (\mathbb{R}_{(-)}^{+}\setminus \{+\infty\})\bigr) \rtimes_{\alpha} \mathbb{R}_d.
\end{equation}
\tag{8.5}
$$
Similarly, for $Q=Q_2(-\infty)$ we obtain an isomorphism
$$
\begin{equation}
\widetilde{\mathcal{J}}_{Q_2(-\infty)} \rtimes_{\alpha^{Q_2(-\infty)}}\mathbb{R}_d \cong C_0\bigl((\mathbb{R}_{(+)}^{-} \setminus\{-\infty\})\cup (\mathbb{R}_{(-)}^{-}\setminus \{+\infty\})\bigr) \rtimes_{\alpha} \mathbb{R}_d.
\end{equation}
\tag{8.6}
$$
For $Q=Q_3(\phi,\psi)$ it follows from (6.5)-(6.6) that the quasi-orbit consists of a single point and is closed. Therefore, $\mathcal{J}_{Q_3(\phi,\psi)}= \mathcal{B}_\infty$ and $\mathcal{I}_{Q_3(\phi,\pm\infty)}=\ker \pi(\phi, \mp,\infty)$; hence $\mathcal{A}_{Q_3(\phi,\psi)}=\widetilde{\mathcal{J}}_{Q_3(\phi,\psi)}\cong \mathbb{C}$. Thus,
$$
\begin{equation}
\widetilde{\mathcal{J}}_{Q_3(\phi,\psi)} \rtimes_{\alpha^{Q_3(\phi,\psi)}}\mathbb{R}_d \cong \mathbb{C} \rtimes_{\alpha}\mathbb{R}_d.
\end{equation}
\tag{8.7}
$$
Since the action on $\mathbb{C}$ is trivial, we have $\mathbb{C} \rtimes_{\alpha}\mathbb{R}_d \cong \mathbb{C}\otimes C^*(\mathbb{R}_d) \cong C^*(\mathbb{R}_d)$ (see [17], Lemma 2.73), where $C^*(\mathbb{R}_d)$ is the group $C^*$-algebra. It is known from the theory of almost periodic functions that $C^*(\mathbb{R}_d)\cong \operatorname{AP}(\mathbb{R})$. Therefore,
$$
\begin{equation}
\widetilde{\mathcal{J}}_{Q_3(\phi,\psi)} \rtimes_{\alpha^{Q_3(\phi,\psi)}}\mathbb{R}_d \cong \operatorname{AP}(\mathbb{R}).
\end{equation}
\tag{8.8}
$$
Now we describe the primitive spectrum $\operatorname{Prim}\mathfrak{T}_\infty$. Notice that for $Q=Q_1(\lambda_0)$ and $Q=Q_2(\pm \infty)$ the algebras $\widetilde{\mathcal{J}}_Q \rtimes_{\alpha^Q}\mathbb{R}_d$ are simple. Indeed, for any $Q$ the quasi-orbit space of $(\widetilde{\mathcal{J}}_Q,\mathbb{R}_d,\alpha^Q)$ consists of a single point, so $\widetilde{\mathcal{J}}_Q $ does not contain nontrivial invariant ideals. Arguing as in the proof of Lemma 4, it is easy to verify that $\mathbb{R}_d$ acts topologically freely on $\widetilde{\mathcal{J}}_Q$ for $Q=Q_1(\lambda_0)$ and $Q=Q_2(\pm \infty)$. According to the corollary on p. 122 of [19], the reduced crossed product $\widetilde{\mathcal{J}}_Q \rtimes_{r,\alpha^Q}\mathbb{R}_d$ is a simple algebra. Since the group $\mathbb{R}_d$ is commutative and therefore amenable, the reduced crossed product coincides with the full one. Every simple algebra $\widetilde{\mathcal{J}}_Q \rtimes_{\alpha^Q}\mathbb{R}_d$ corresponds to a unique primitive ideal ${\mathcal{I}_Q \rtimes_{\alpha^Q}\mathbb{R}_d}$ of $\mathcal{B}_\infty \rtimes_{\alpha^Q}\mathbb{R}_d$. Indeed, since the algebra $\widetilde{\mathcal{J}}_Q \rtimes_{\alpha^Q}\mathbb{R}_d$ is simple, any irreducible representations $\pi''\rtimes U$ of it is faithful. The extension of such a representation to an irreducible representation of $\mathcal{B}_\infty \rtimes_{\alpha^Q}\mathbb{R}_d$ is of the form $\pi\rtimes U$, where $U$ is faithful and $\ker \pi=\mathcal{I}_Q$ (see the proof of Theorem 2). Therefore, $\ker( \pi \rtimes U)=\mathcal{I}_Q \rtimes_{\alpha^Q}\mathbb{R}_d$. For $Q=Q_3(\phi,\psi)$ every irreducible representation of $\widetilde{\mathcal{J}}_{Q} \rtimes_{\alpha^{Q}}\mathbb{R}_d$ has the form $\operatorname{id}\rtimes \zeta$ (see equality (8.7)), where $\zeta \in \operatorname{Hom}(\mathbb{R}_d,U(1))$ is a character of the group $\mathbb{R}_d$. The set of characters of $\mathbb{R}_d$ equipped with the compact-open topology is called the Bohr compactification $b\mathbb{R}$ of the group of real numbers. We note that a character $\zeta$ of $\mathbb{R}_d$ extends naturally to a representation $\pi_\zeta$ of the algebra $\operatorname{AP}(\mathbb{R})$ by the rule and the space $b\mathbb{R}$ is homeomorphic to $\widehat{\operatorname{AP}(\mathbb{R})}$. The extension of $\operatorname{id}\rtimes \zeta$ to an irreducible representation of $\mathcal{B}_\infty \rtimes_{\alpha}\mathbb{R}_d$ is of the form $\pi\rtimes \zeta$, where $\ker \pi=\mathcal{I}_{Q_3}(\phi,\psi)$. We define an embedding $\ker \pi_\zeta \to \mathcal{B}_\infty \rtimes_{\alpha}\mathbb{R}_d$ by $f \mapsto [f]_\infty$. Then $\ker (\pi \rtimes \zeta)$ is the ideal generated by $\mathcal{I}_{Q_3}(\phi,\psi) \rtimes_{\alpha} \mathbb{R}_d$ and the image of $\ker \pi_\zeta$ under the embedding into $\mathcal{B}_\infty\rtimes_{\alpha}\mathbb{R}_d$.To describe the topology on $\operatorname{Prim}\mathfrak{T}_\infty$ it is convenient to introduce an auxiliary topological space. Consider the disjoint union $\Omega=\mathcal{Q}_{1,2} \cup (\mathcal{Q}_{3} \times b\mathbb{R})$, where $\mathcal{Q}_{1,2}=\{Q_1(\lambda)\colon \lambda \in \mathbb{R}\} \cup \{ Q_2(\phi)\colon \phi=\pm \infty \}$ and $\mathcal{Q}_3=\mathcal{Q}\setminus \mathcal{Q}_{1,2}$. We introduce a fundamental system of neighbourhoods on $\Omega$. The system of neighbourhoods of the points $Q=Q_1(\lambda)$ consists of sets of the form (6.7) and, for $Q=Q_2(\phi)$ it consists of the sets (6.8). For a point $(Q,\zeta) \in \mathcal{Q}_3\times b\mathbb{R}$ the neighbourhood system consists of the sets $(V_Q\cap \mathcal{Q}_{1,2}) \cup (\{Q\}\times V_f)\subset \Omega$, where $V_Q$ has the form (6.9) and $V_f$ is a neighbourhood of the point $\zeta \in b\mathbb{R}$ of the form
$$
\begin{equation}
V_f=\bigl\{\eta \in b\mathbb{R}\colon \| \pi_\eta(f) \| > 0 \bigr\}
\end{equation}
\tag{8.10}
$$
for fixed $f\in \operatorname{AP}(\mathbb{R})$. We define a bijection $\Psi\colon \Omega \to \operatorname{Prim} \mathfrak{T}_\infty$ by
$$
\begin{equation}
Q_1(\lambda) \mapsto \mathcal{I}_{Q_1(\lambda)}\rtimes_{\alpha} \mathbb{R}_d,
\end{equation}
\tag{8.11}
$$
$$
\begin{equation}
Q_2(\phi) \mapsto \mathcal{I}_{Q_2(\phi)}\rtimes_{\alpha} \mathbb{R}_d
\end{equation}
\tag{8.12}
$$
and
$$
\begin{equation}
(Q_3(\phi,\psi),\zeta) \mapsto \mathcal{I}_{Q_3(\phi,\psi),\zeta},
\end{equation}
\tag{8.13}
$$
where $\mathcal{I}_{Q_3(\phi,\psi),\zeta}$ is the ideal generated by $\mathcal{I}_{Q_3(\phi,\psi)}\rtimes_{\alpha}\mathbb{R}_d$ and $\ker \pi_\zeta$. Let us show that $\Psi$ is a homeomorphism. First we prove an auxiliary lemma. Lemma 9. Let $(\mathcal{A},G,\alpha)$ be a dynamical system with discrete abelian1 group $G$. Assume that the action of $G$ is free2 on $\widehat{\mathcal{A}}$, and the quasi-orbit space $\mathcal{Q}$ is almost Hausdorff. Then the map $\mathcal{Q}\to \operatorname{Prim}\mathcal{A}\rtimes_{\alpha}G\colon Q \mapsto \mathcal{I}_Q\rtimes_{\alpha}G$ is a homeomorphism. Proof. It suffices to establish that each map in the following chain is a homeomorphism:
$$
\begin{equation*}
\begin{aligned} \, & Q\mapsto \mathcal{I}_Q\mapsto \mathcal{I}_Q \mapsto \mathcal{I}_Q\rtimes_{r,\alpha} G \mapsto \mathcal{I}_Q\rtimes_{r,\alpha}G \mapsto \mathcal{I}_Q\rtimes_{\alpha}G\colon \\ &\quad \mathcal{Q} \approx \operatorname{Prim}^G \mathcal{A}=\operatorname{Prime}^G \mathcal{A} \approx \operatorname{Prime} \mathcal{A}\rtimes_{r,\alpha} G=\operatorname{Prim} \mathcal{A}\rtimes_{r,\alpha} G \approx \operatorname{Prim} \mathcal{A}\rtimes_{\alpha} G, \end{aligned}
\end{equation*}
\notag
$$
where $\operatorname{Prim}^G \mathcal{A} $ and $\operatorname{Prime}^G \mathcal{A} $ are the sets of $G$-primitive and $G$-prime ideals of $\mathcal{A}$, respectively (the definitions are given in [23], before Lemma 2.4 and Proposition 2.3), and $\operatorname{Prime} \mathcal{A}\rtimes_{r,\alpha} G$ is the set of prime ideals of $\mathcal{A}\rtimes_{r,\alpha} G$; all three sets mentioned above are equipped with the Fell topology (see the definition before Theorem 2.1 in [23]). From the lemma on p. 221 in [24] it follows that the map $\mathcal{Q} \to \operatorname{Prim}^G \mathcal{A}\colon Q \mapsto \mathcal{I}_Q$ is a homeomorphism. The equality $\operatorname{Prim}^G \mathcal{A}=\operatorname{Prime}^G \mathcal{A}$ was derived in the proof of Lemma 2.5 in [23] from the condition that each closed irreducible subset of $\operatorname{Prim}^G \mathcal{A}$ is the closure of a point. This property follows from the almost Hausdorff property of $\operatorname{Prim}^G \mathcal{A}$ (the lemma on p. 222 in [24]), which, in turn, follows from the almost Hausdorff property of $\mathcal{Q}$ and the existence of a homeomorphism $\mathcal{Q} \approx \operatorname{Prim}^G \mathcal{A}$. The map $\operatorname{Prime}^G \mathcal{A} \to \operatorname{Prime} \mathcal{A}\rtimes_{r,\alpha} G\colon \mathcal{I}_Q \mapsto \mathcal{I}_Q\rtimes_{r,\alpha}G$ is a homeomorphism by Lemma 2.3 in [23], since the action of $G$ is free on $\widehat{\mathcal{A}}$. In particular, this means that $\operatorname{Prime} \mathcal{A}\rtimes_{r,\alpha} G$ is an almost Hausdorff space. The primitive spectrum $\operatorname{Prim} \mathcal{A}\rtimes_{r,\alpha} G$ is obviously a subset of $\operatorname{Prime} \mathcal{A}\rtimes_{r,\alpha} G$, and the restriction of the Fell topology to $\operatorname{Prim} \mathcal{A}\rtimes_{r,\alpha} G$ coincides with the Jacobson topology. Since $\operatorname{Prim} \mathcal{A}\rtimes_{r,\alpha} G$ is a locally compact space and $\operatorname{Prime} \mathcal{A}\rtimes_{r,\alpha} G$ is almost Hausdorff, $\operatorname{Prim} \mathcal{A}\rtimes_{r,\alpha} G$ is almost Hausdorff too. Hence, similarly to the equality $\operatorname{Prim}^G \mathcal{A}=\operatorname{Prime}^G \mathcal{A}$, we derive $\operatorname{Prime} \mathcal{A}\rtimes_{r,\alpha} G=\operatorname{Prim} \mathcal{A}\rtimes_{r,\alpha} G$. Finally, since $G$ is abelian, we have $\mathcal{J}\rtimes_{r,\alpha} G \cong \mathcal{J}\rtimes_{\alpha} G$ for any ideal $\mathcal{J}\subset \mathcal{A}$, which implies the existence of the last homeomorphism.
The lemma is proved. Recall that for any $C^*$-algebra $\mathcal{A}$ the Jacobson topology on $\operatorname{Prim} \mathcal{A}$ is generated by the basis of neighbourhoods $\{U_a\colon a \in \mathcal{A}\}$, where
$$
\begin{equation}
U_a=\bigl\{\mathcal{J} \in \operatorname{Prim}\mathcal{A}\colon \| a \|_{\mathcal{J}} > 0 \bigr\},
\end{equation}
\tag{8.14}
$$
and $\|a\|_{\mathcal{J}}=\inf_{j\in \mathcal{J}} \| a+j \|$. Below we describe the topology on $\operatorname{Prim}\mathfrak{T}_\infty$; for this purpose we need some auxiliary definitions. Let $(\mathcal{A},G,\alpha)$ be a dynamical system. Consider the map $\widetilde{E}\colon C_c(G,\mathcal{A}) \to \mathcal{A}\colon f \mapsto f(0)$. From [25], Theorem 4.1, it follows that $\widetilde{E}$ extends to a bounded positive map
$$
\begin{equation}
E\colon \mathcal{A}\rtimes_{\alpha}G \to \mathcal{A},
\end{equation}
\tag{8.15}
$$
where $\| E \|=1$. If the reduced crossed product $\mathcal{A}\rtimes_{\alpha,r}G$ coincides with $\mathcal{A}\rtimes_{\alpha}G$, then it follows from Proposition 4.1.9 in [26] that if $a \geqslant 0$ and $E(a)=0$, then $a=0$. Let $\mathcal{I}$ be an invariant ideal of $\mathcal{A}$. We denote by $E_\mathcal{I}$ the map defined by (8.15) for $\mathcal{A}$ replaced by $\mathcal{A}/\mathcal{I}$. For $\mathcal{I}=\{0\}$ we still write $E$ instead of $E_{\mathcal{I}}$. Let $\rho_\mathcal{I}\colon \mathcal{A}\to \mathcal{A}/ \mathcal{I}$ be the quotient map, and let $p_\mathcal{I}\colon \mathcal{A} \rtimes_{\alpha} G \to (\mathcal{A}/\mathcal{I}) \rtimes_{\alpha^\mathcal{I}}G$ be the $*$-homomorphism obtained by extending the $*$-homomorphism $C_c(G,\mathcal{A}) \to C_c(G,\mathcal{A}/\mathcal{I})\colon f(\,\cdot\,) \mapsto \rho_\mathcal{I}(f(\,\cdot\,))$ by continuity. Then it is obvious that the relation $\rho_\mathcal{I} \circ E=E_\mathcal{I}\circ p_\mathcal{I}$ holds. Theorem 3. The topology on $\operatorname{Prim} \mathfrak{T}_\infty$ induced by $\Psi$ coincides with the Jacobson topology. Proof. First we consider the restriction of $\Psi$ to $\mathcal{Q}_{1,2}$. Set $\mathcal{J}=\bigcap_{P \in \kappa^{-1}(\mathcal{Q}_3)}P$. Clearly, $\mathcal{J}$ is an invariant ideal of $\mathcal{B}_\infty$, its spectrum $\operatorname{Prim}\mathcal{J}$ coincides with $\kappa^{-1}(\mkern-1mu\mathcal{Q}_{1,2}\mkern-1mu)$, the group $\mathbb{R}_d$ acts freely on $\operatorname{Prim}\mathcal{J}$, and the quasi-orbit space $\mathcal{Q}_{1,2}$ of the dynamical system $(\mathcal{J},\mathbb{R}_d,\alpha)$ is almost Hausdorff (see the argument before Lemma 6). According to Lemma 9 (for $\mathcal{A}=\mathcal{J}$ and $G=\mathbb{R}_d$), the map $\Psi$ implements a homeomorphism between the open subsets $\mathcal{Q}_{1,2} \subset \Omega$ and $\operatorname{Prim}\mathcal{J}\rtimes_{\alpha}\mathbb{R}_d \subset \operatorname{Prim} \mathfrak{T}_\infty$. To complete the proof it remains to compare the bases of neighbourhoods $U_a$ and $\Psi(V)$, where $V=(V_Q\cap \mathcal{Q}_{1,2}) \cup (\{Q\}\times V_f)$, for ideals $\mathcal{I}_{Q,\zeta}$ such that $Q=Q_3(\phi,\psi)$.
Let $U_a$ be a neighbourhood of the form (8.14) in $\operatorname{Prim} \mathfrak{T}_\infty$ containing the ideal $\mathcal{I}_{Q,\zeta}$. We will construct a neighbourhood $V$ of the form $(V_Q\cap \mathcal{Q}_{1,2}) \cup (\{Q\}\times V_f)\subset \Omega$, such that $\mathcal{I}_{Q,\zeta} \in \Psi (V) \subset U_a$. From $\| a^*a \|=\|a\|^2$ it follows that $U_a=U_{a^*a}$, and therefore we can assume that $a$ is a positive nonzero element. As $a \geqslant 0$, we have $p_{\mathcal{I}_Q}(a) \geqslant 0$ (because $p_{\mathcal{I}_Q}$ is a $*$-homomorphism; see the definition of $p_{\mathcal{I}_Q}$ before the present theorem). Recall that $\mathcal{I}_{Q,\zeta}$ is generated by the sets $\mathcal{I}_{Q}\rtimes_{\alpha}\mathbb{R}_d$ and $\ker \pi_\zeta$. The inclusion $\mathcal{I}_{Q} \rtimes \mathbb{R}_d \subset \mathcal{I}_{Q,\zeta}$ implies that $\|a\|_{\mathcal{I}_{Q} \rtimes_{\alpha} \mathbb{R}_d} \geqslant \|a\|_{\mathcal{I}_{Q,\zeta}}$. Since $\mathcal{I}_{Q,\zeta} \in U_a$, the norm $\|a\|_{\mathcal{I}_{Q,\zeta}}$ is nonzero, as also is the norm $\|a\|_{\mathcal{I}_Q \rtimes_{\alpha}\mathbb{R}_d}$. According to [17], Proposition 3.19, we have $\ker p_{\mathcal{I}_Q}=\mathcal{I}_Q\rtimes_{\alpha}\mathbb{R}_d$, and so $\|a\|_{\mathcal{I}_Q \rtimes_{\alpha}\mathbb{R}_d} =\| p_{\mathcal{I}_Q}(a) \| >0$. Let $b=E(a)$; then $\rho_{\mathcal{I}_{Q}}(b)=\rho_{\mathcal{I}_{Q}}(E(a))= E_{\mathcal{I}_{Q}}(p_{\mathcal{I}_{Q}}(a))$. Since $p_{\mathcal{I}_Q}(a)$ is positive and nonzero, we have $ E_{\mathcal{I}_{Q}}(p_{\mathcal{I}_{Q}}(a)) \neq 0$, and thus $\| b \|_{\mathcal{I}_{Q}}=\| \rho_{\mathcal{I}_Q}(b) \| > 0$. Therefore, the ideal $\mathcal{I}_{Q}$ belongs to the set $V^0_b=\{\mathcal{J} \in \operatorname{Prim} \mathcal{B}_\infty\colon \| b \|_{\mathcal{J}}>0 \}$, which is open in the Jacobson topology on $\operatorname{Prim}\mathcal{B}_\infty$. By Lemma 3 there exists a neighbourhood $V_{b}$ in $\operatorname{Prim}\mathcal{B}_\infty$ which has one of the forms 2–5 introduced before Lemma 3 (the form of this neighbourhood depends on $\xi=\psi$ and $\omega=\operatorname{sgn} \phi$), such that Let $V_Q=\kappa(V_b)$, so $V_Q$ is a neighbourhood of the form (6.9) in $\mathcal{Q}$. Let $f \in AP(\mathcal{\mathbb{R}})$ be the image of $a$ under the map $p_{\mathcal{I}_Q}\colon \mathcal{B}_\infty \rtimes_{\alpha} \mathbb{R}_d \to \widetilde{\mathcal{J}_Q}\rtimes_{\alpha^Q}\mathbb{R}_d=\operatorname{AP}(\mathbb{R})$ (see (8.8)). Finally, let Since $\zeta \in V_f$, we have $\mathcal{I}_{Q,\zeta} \in \Psi(\{Q\}\times V_f)\subset \Psi (V)$.Let us show that $\Psi (V) \subset U_a$. Let $\mathcal{J} \in \Psi (V_Q\cap \mathcal{Q}_{1,2})$, that is, $\mathcal{J}=\mathcal{I}_{Q'}\rtimes_{\alpha} \mathbb{R}_d$ for some $Q'\in V_Q \cap \mathcal{Q}_{1,2}$. From $V_Q=\kappa(V_b)$ it follows that the set $\kappa^{-1}(Q')\cap V_b$ is nonempty. From this and the inclusion $V_b \subset V_b^0$ we conclude that there exists an ideal $\mathcal{I} \in \kappa^{-1}(Q')$ such that $ \| b \|_{\mathcal{I}} > 0 $. Therefore,
$$
\begin{equation*}
\|a\|_{\mathcal{I}_{Q'}\rtimes_{\alpha}\mathbb{R}_d} =\| p_{\mathcal{I}_{Q'}}(a) \| \geqslant \| E_{\mathcal{I}_{Q'}} ( p_{\mathcal{I}_{Q'}}(a)) \|=\| b \|_{\mathcal{I}_{Q'}} \geqslant \| b \|_{\mathcal{I}}>0,
\end{equation*}
\notag
$$
that is, $\mathcal{J}=\mathcal{I}_{Q'}\rtimes_{\alpha} \mathbb{R}_d \in U_a$. If $\mathcal{J} \in \Psi(\{Q\}\times V_f)$, then $\mathcal{J}=\mathcal{I}_{Q,\eta}$. Then the inclusion $\mathcal{J} \in U_a$ follows from the equalities $\|a\|_{\mathcal{J}}=\| p_{\mathcal{I}_Q}(a) \|_{\ker \pi_\eta}=\| \pi_\eta (p_{\mathcal{I}_Q}(a) )\|=\| \pi_\eta(f) \|$ and from the inclusion $ \eta \in V_f$.
Conversely, consider a neighbourhood $V=(V_Q\cap \mathcal{Q}_{1,2}) \cup (\{Q\}\times V_f)$ of a point $(Q,\zeta)\in \mathcal{Q}_3\times b\mathbb{R}$, where $V_Q$ is a neighbourhood in $\mathcal{Q}$ of $Q$ of the form (6.9) and $V_f$ is a neighbourhood of $\zeta$ in $b\mathbb{R}$ of the form (8.10) for some $f \in \operatorname{AP}(\mathbb{R})$. We construct an element $a\in \mathcal{B}_\infty \rtimes_{\alpha}\mathbb{R}_d$ such that $\mathcal{I}_{Q,\zeta} \in U_a \subset \Psi(V)$. Recall that for $Q \in \mathcal{Q}_{3}$, the set $\kappa^{-1}(Q)$ consists of the single element $\mathcal{I}_Q$. The set $\kappa^{-1}(V_Q)\subset \operatorname{Prim}\mathcal{B}_\infty$ is open in the Jacobson topology; thus, $\kappa^{-1}(Q)$ is contained in it along with an open neighbourhood of the form (8.14). In other words, there exists $b \in \mathcal{B}_\infty $ such that the following inclusions hold:
$$
\begin{equation}
\kappa^{-1}(Q) \subset V^0_b=\bigl\{\mathcal{I}\in \operatorname{Prim}\mathcal{B}_\infty\colon \| b \|_\mathcal{I}>0 \bigr\} \subset \kappa^{-1}(V_Q).
\end{equation}
\tag{8.16}
$$
Let $a=b \cdot [f]_\infty$. Since $\widetilde{\mathcal{J}}_Q \rtimes_{\alpha^Q}\mathbb{R}_d \cong \mathbb{C}\otimes \operatorname{AP}(\mathbb{R})$ (see the discussion after (8.7)) and $p_{\mathcal{I}_Q}(a)=\rho_{\mathcal{I}_Q}(b) \otimes f $, we have $\|a\|_{\mathcal{I}_{Q,\zeta}}=\| p_{\mathcal{I}_Q}(a) \|_{\ker \pi_\zeta}= \| b \|_{\mathcal{I}_Q} \| \pi_\zeta(f) \|\,{>}\,0$. Therefore, we have the inclusion $\mathcal{I}_{Q,\zeta} \in U_a$.
Now we show that $U_a \subset \Psi(V)$. First, notice that the inequality $\|a\|_{\mathcal{J}} \leqslant \| b \|_{\mathcal{J}} \cdot \| [f]_\infty \|_{\mathcal{J}}$ implies that $U_a \subset U_b\cap U_{[f]_\infty}$. Obviously, $U_{[f]_\infty} \subset \Psi(\mathcal{Q}_{1,2}\cup (\mathcal{Q}_3\times V_f))$. Additionally, we establish below the relation
$$
\begin{equation}
U_b \subset \Psi \bigl( (V_Q\cap \mathcal{Q}_{1,2})\cup (\{Q\}\times b\mathbb{R})\bigr).
\end{equation}
\tag{8.17}
$$
Thus,
$$
\begin{equation*}
U_a \subset U_b\cap U_{[f]_\infty}=\Psi \bigl( (V_Q\cap \mathcal{Q}_{1,2})\cup (\{Q\}\times b\mathbb{R})\bigr) \cap \Psi\bigl(\mathcal{Q}_{1,2}\cup (\mathcal{Q}_3\times V_f)\bigr)= \Psi(V),
\end{equation*}
\notag
$$
as required.
Let us verify (8.17). Suppose that $\mathcal{J}\in U_b$ and $Q'$ is the quasi-orbit associated with $\mathcal{J}$. There exists an ideal $\mathcal{I} \in \kappa^{-1}(Q')$ such that $\| b \|_{\mathcal{I}} > 0$. Otherwise we would have $b \in \mathcal{I}$ for each $\mathcal{I}\in \kappa^{-1}(Q')$, which leads to the inclusion $b \in \mathcal{I}_{Q'}=\bigcap_{\mathcal{I}\in \kappa^{-1}(Q')} \mathcal{I}$, that is, $\| b \|_{\mathcal{I}_{Q'}}=0 $. Then it follows from the inequality ${\| b \|_{\mathcal{J}} \leqslant \| b \|_{\mathcal{I}_{Q'}}}$ that $\| b \|_{\mathcal{J}}=0$, which contradicts the inclusion $\mathcal{J} \in U_b$. Thus, $\mathcal{I} \in V_b^0$. The element $b$ was chosen in such a way that $\kappa (V_b^0) \subset V_Q$: see (8.16); therefore, we have $Q'=\kappa (\mathcal{I}) \in V_Q$. There are two possibilities: either $Q' \in V_Q \cap \mathcal{Q}_{1,2}$, and then ${\mathcal{J}=\mathcal{I}_{Q'}\rtimes_{\alpha}\mathbb{R}_d}$, or $Q' \in V_Q \cap \mathcal{Q}_3$, in which case $\mathcal{J}= \mathcal{I}_{Q',\eta}$ for some $\eta \in b \mathbb{R}$ (see (8.11)–(8.13)). In other words we have the relation
$$
\begin{equation*}
\mathcal{J} \in \Psi \bigl( (V_Q\cap \mathcal{Q}_{1,2})\cup ((V_Q\cap\mathcal{Q}_3)\times b\mathbb{R})\bigr).
\end{equation*}
\notag
$$
From this and the equality $V_Q \cap \mathcal{Q}_3=\{Q\}$ we obtain (8.17).
The theorem is proved. § 9. Primitive ideals in $\mathfrak{T}$First we present a list of irreducible representations of $\mathfrak{T}$ whose kernels make up $\operatorname{Prim} \mathfrak{T}$. Note that the classes of representations in this list do not exhaust the entire spectrum $\widehat{\mathfrak{T}}$. As mentioned before (see Remark 1), the union on the right-hand side of the equality
$$
\begin{equation*}
\widehat{\mathfrak{T}}=\bigcup_{x\in \dot{\mathbb{R}}} \widehat{\mathfrak{T}_{x}} \cup [\operatorname{Id}]
\end{equation*}
\notag
$$
is disjoint. Therefore, it suffices to present a similar list for each local algebra $\mathfrak{T}_x$ and then extend these representations to the initial algebra $\mathfrak{T}$. For $x_0\neq \infty$ the spectrum $\widehat{\mathfrak{T}_{x_0}}$ consists of two classes corresponding to the irreducible representations (2.4). These representations extend to irreducible representations $\pi(x_0,\pm)$ of $\mathfrak{T}$; their action is given by Let $x_0=\infty$. Recall the equality (see Theorem 2)
$$
\begin{equation*}
\widehat{\mathfrak{T}_\infty}=\bigcup_{Q \in \mathcal{Q}} (\widetilde{\mathcal{J}}_Q \rtimes_{\alpha^Q}\mathbb{R}_d)\,\widehat{}\ ,
\end{equation*}
\notag
$$
where $\widetilde{\mathcal{J}}_Q=\mathcal{J}_Q/\mathcal{I}_Q$ and the union on the right-hand side is disjoint. The primitive spectrum of each algebra $\widetilde{\mathcal{J}}_Q \rtimes_{\alpha^Q}\mathbb{R}_d$ was described in § 8. With each primitive ideal $\mathcal{I}$ in $\widetilde{\mathcal{J}}_Q \rtimes_{\alpha^Q}\mathbb{R}_d$ we associate an irreducible representation whose kernel coincides with $\mathcal{I}$. For $Q=Q_1(\lambda_0)$, $\lambda_0 \in \mathbb{R}$, there is a unique primitive ideal in $\widetilde{\mathcal{J}}_Q \rtimes_{\alpha^Q}\mathbb{R}_d$. Consider the representations $\pi''\colon \widetilde{\mathcal{J}}_{Q_1(\lambda_0)}\to B(L_2(\mathbb{R}_d)\oplus L_2(\mathbb{R}_d) )$ and $U\colon \mathbb{R}_d \to B(L_2(\mathbb{R}_d)\oplus L_2(\mathbb{R}_d))$ defined by
$$
\begin{equation*}
\begin{gathered} \, (\pi''([b]_\infty+\mathcal{I}_{Q_1(\lambda_0)})w) (\xi) = \pi(\xi,\lambda_0)([b]_\infty)w(\xi), \\ (U(k)w)(\xi)=w(\xi+k), \end{gathered}
\end{equation*}
\notag
$$
where $\pi(\xi,\lambda_0)$ is the representation from (4.6)–(4.7), $w \in L_2(\mathbb{R}_d)\oplus L_2(\mathbb{R}_d)$ and ${k \,{\in}\, \mathbb{R}_d}$. It is clear that $(\pi'',U)$ is a covariant representation of $(\widetilde{\mathcal{J}}_{Q_1(\lambda_0)}, \mathbb{R}_d,\alpha^{Q_1(\lambda_0)})$. The map $\pi'' \rtimes U$ implements an irreducible representation of $\widetilde{\mathcal{J}}_{Q_1(\lambda_0)} \rtimes_{\alpha^{Q_1(\lambda_0)}}\mathbb{R}_d$. Otherwise, there would exist a nontrivial closed subspace $\mathcal{F} \subset L_2(\mathbb{R}_d)\oplus L_2(\mathbb{R}_d)$ such that $\pi''([b]_\infty+\mathcal{I}_{Q_1(\lambda_0)})\mathcal{F}\subset \mathcal{F}$ and $U(k)\mathcal{F}\subset \mathcal{F}$ for all $[b]_\infty \in \mathcal{J}_{Q_1(\lambda_0)}$, $k \in \mathbb{R}_d$ (see the discussion before Lemma 7). From the inclusion $\pi''([b]_\infty+\mathcal{I}_{Q_1(\lambda_0)})\mathcal{F}\subset \mathcal{F}$ it follows that $\mathcal{F}=\chi_E (L_2(\mathbb{R}_d)\oplus L_2(\mathbb{R}_d))$, where $\chi_E$ is the characteristic function of some nonempty set $E\subset \mathbb{R}_d$. The inclusion $U(k)\mathcal{F}\subset \mathcal{F}$ means that $E$ is invariant under shifts, and therefore $E=\mathbb{R}_d$ and $\mathcal{F}=L_2(\mathbb{R}_d)\oplus L_2(\mathbb{R}_d)$, which is a contradiction. By (8.4) every irreducible representation of $\widetilde{\mathcal{J}}_{Q_1(\lambda_0)} \rtimes_{\alpha^{Q_1(\lambda_0)}}\mathbb{R}_d$ is equivalent to ${\pi''\rtimes U}$. There exists a unique extension of the representation $\pi''$ of $\widetilde{\mathcal{J}}_{Q_1(\lambda_0)} =\mathcal{J}_{Q_1(\lambda_0)} / \mathcal{I}_{Q_1(\lambda_0)}$ to a representation $\pi'$ of $\mathcal{B}_\infty/ \mathcal{I}_{Q_1(\lambda_0)}$ on the same Hilbert space. The representation $\pi'$, in turn, extends uniquely to a representation of $\mathcal{B}_\infty$. It can easily be seen that the representation $\pi$ defined by
$$
\begin{equation*}
(\pi([b]_\infty)w) (\xi)=\pi(\xi,\lambda_0)([b]_\infty)w(\xi),
\end{equation*}
\notag
$$
where $w \in L_2(\mathbb{R}_d)\oplus L_2(\mathbb{R}_d)$, implements such an extension. The pair $(\pi,U)$ is a covariant representation of $(\mathcal{B}_\infty, \mathbb{R}_d,\alpha)$. The representation $\pi \rtimes U$ of $\mathcal{B}_\infty \rtimes_{\alpha}\mathbb{R}_d$ is irreducible, since it is an extension of the irreducible representation $\pi''\rtimes U$. There exists a unique extension $\pi(\lambda_0)$ of $\pi \rtimes U$ to $\mathfrak{T}$ on the same Hilbert space $L_2(\mathbb{R}_d)\oplus L_2(\mathbb{R}_d)$. Let us describe the action of $\pi(\lambda_0)$ on elements of the form $aI+bS$ for $a,b \in \operatorname{SAP}(\mathbb{R})$. First consider the action of $\pi(\lambda_0)$ on $a \in \operatorname{SAP}(\mathbb{R})$. Using the decomposition (2.1) we obtain
$$
\begin{equation*}
\begin{aligned} \, \pi(\lambda_0)(a) &=\pi \rtimes U ([a]_\infty) \\ &= \pi ([\chi_+]_\infty) \cdot \pi\rtimes U ([a_+]_\infty)+\pi ([\chi_-]_\infty) \cdot \pi\rtimes U ([a_-]_\infty), \end{aligned}
\end{equation*}
\notag
$$
where we take into account the obvious equalities $[a_0]_\infty=0$ and $\pi \rtimes U ([\chi_\pm]_\infty)=\pi ([\chi_\pm]_\infty)$. From equality (4.6) it follows that
$$
\begin{equation*}
\pi([\chi_+]_\infty)=\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}\quad\text{and} \quad \pi([\chi_-]_\infty)=\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}.
\end{equation*}
\notag
$$
Let $f_N=\sum_{i=1}^N c_i e_{k_i}$ be a trigonometric polynomial. Then
$$
\begin{equation}
\pi \rtimes U ([f_N]_\infty)=\sum_{i=1}^N c_i U(k_i).
\end{equation}
\tag{9.2}
$$
We denote a shift operator in the space $L_2(\mathbb{R}_d)$ by $T_{k_i}$: $(T_{k_i}v)(\xi)\,{=}\,v(\xi+ k_i)$, and let $T_{f_N}=\sum_{i=1}^N c_i T_{k_i}$. Then formula (9.2) takes the form
$$
\begin{equation*}
\pi \rtimes U ([f_N]_\infty)= \begin{pmatrix} T_{f_N} & 0 \\ 0 & T_{f_N} \end{pmatrix}.
\end{equation*}
\notag
$$
From this and the equality $\| \pi \rtimes U \|=1 $ it follows that $\| T_{f_N} \| \leqslant \| f_N \| $. We approximate $f \in \operatorname{AP}(\mathbb{R})$ by trigonometric polynomials $f_N$, and let $T_f=\lim_{N\to \infty} T_{f_N}$. Thus,
$$
\begin{equation*}
\pi(\lambda_0)(a)=\begin{pmatrix} T_{a_+} & 0 \\ 0 & T_{a_-} \end{pmatrix}.
\end{equation*}
\notag
$$
Further, the relation $\pi(\lambda_0)(S)=\pi ([S]_\infty)$ holds. Let $h_0$ be the operator of multiplication by the function $h_0(\xi)= \operatorname{sgn} \xi$, where we set the function $\operatorname{sgn}$ to be equal to zero at $\xi=0$, and let $\delta_0$ be the operator of multiplication by the function equal to one at $0$ and zero elsewhere. The equality $S=- F^{-1}h_0 F$ and relation (4.7) lead to the equality
$$
\begin{equation*}
\pi(\lambda_0)(S)=(\delta_0 \cdot B(\lambda_0) -h_0 \cdot I).
\end{equation*}
\notag
$$
Summarizing, we obtain
$$
\begin{equation}
\pi(\lambda_0)(aI+bS)= \begin{pmatrix} T_{a_+} & 0 \\ 0 & T_{a_-} \end{pmatrix}+ \begin{pmatrix} T_{b_+} & 0 \\ 0 & T_{b_-} \end{pmatrix} (\delta_0 \cdot B(\lambda_0) - h_0 \cdot I), \qquad \lambda_0 \in \mathbb{R},
\end{equation}
\tag{9.3}
$$
where $a_\pm, b_\pm \in \operatorname{AP}(\mathbb{R})$ are the almost periodic functions in the decomposition (2.1) for $a,b \in \operatorname{SAP}(\mathbb{R})$. For $Q=Q_2(\phi)$, $\phi=\pm \infty$, the algebra $\widetilde{\mathcal{J}}_Q \rtimes_{\alpha^Q}\mathbb{R}_d$, as in the previous case, has a unique primitive ideal. However, it can be shown that this algebra has infinitely many pairwise inequivalent irreducible representations; we do not describe them here. Consider one such representation $\pi''\rtimes U$ in the Hilbert space $L_2(\mathbb{R})$, where $\pi''$ and $U$ are defined by
$$
\begin{equation*}
\begin{gathered} \, \pi''\colon[a]_\infty+\mathcal{I}_{Q_2(\phi)} \mapsto a(\phi), \qquad a\in C(\overline{\mathbb{R}}), \\ \pi''\colon[F^{-1}h_lF]_\infty+\mathcal{I}_{Q_2(\phi)} \mapsto h_l, \qquad h_l(\xi)= \operatorname{sgn}(\xi+l), \end{gathered}
\end{equation*}
\notag
$$
and where $t_k$ is the shift operator by $k$ in $L_2(\mathbb{R})$, $(t_k v)(\xi)=v(\xi+k)$. Reasoning as in the case $Q=Q_1(\lambda_0)$ it can be verified that such a representation is irreducible, and it has a unique extension $\pi(\phi)$ to the algebra $\mathfrak{T}$. Let us describe its action on the generators $aI+bS$, where $a,b \in \operatorname{SAP}(\mathbb{R})$. For $f \in \operatorname{AP}(\mathbb{R})$, let $t_f=\lim_{N\to \infty} t_{f_N}$, where $f_N \to f$, $f_N=\sum_{i=1}^N c_i e_{k_i}$ is a trigonometric polynomial and $t_{f_N}=\sum_{i=1}^N c_i t_{k_i}$. Then
$$
\begin{equation}
\begin{aligned} \, \pi(\pm \infty)(aI+bS)=t_{a_{\pm}} - t_{b_{\pm}} h_0, \end{aligned}
\end{equation}
\tag{9.4}
$$
where $a_\pm, b_\pm \in \operatorname{AP}(\mathbb{R})$ are almost periodic functions in the decomposition (2.1) for $a,b \in \operatorname{SAP}(\mathbb{R})$ and $h_0 $ is the operator of multiplication by the function $\operatorname{sgn} \xi$. For each quasi-orbit $Q= Q_3(\phi,\psi)$, $\phi,\psi=\pm \infty$, there is an associated series of pairwise inequivalent irreducible representations $\pi''\rtimes \zeta $ of the algebra $\widetilde{\mathcal{J}}_{Q_3(\phi,\psi)} \rtimes_{\alpha^{Q_3(\phi,\psi)}}\mathbb{R}_d$, where
$$
\begin{equation*}
\pi''([b]_\infty+\mathcal{I}_{Q_3(\phi,\pm \infty)})=\pi(\phi,\mp, \pm \infty)([b]_\infty),
\end{equation*}
\notag
$$
$\pi(\phi,\mp, \pm \infty)$ is defined by equations (4.3) and (4.5), and $\zeta \in b\mathbb{R}$ is a character of the group $\mathbb{R}_d$. We denote by $\pi(\phi,\psi,\zeta)$ the extensions of such representations to irreducible representations of the algebra $\mathfrak{T}$. The representations $\pi(\phi,\psi,\zeta)$ act on the generators by the rule
$$
\begin{equation}
\begin{aligned} \, \pi(\pm \infty,\psi,\zeta)( aI+bS )= \pi_\zeta(a_\pm) - \pi_\zeta(b_\pm) \operatorname{sgn} \psi, \end{aligned}
\end{equation}
\tag{9.5}
$$
where $a_\pm, b_\pm \in \operatorname{AP}(\mathbb{R})$ are the almost periodic functions in the decomposition (2.1) for $a,b \in \operatorname{SAP}(\mathbb{R})$ and $\pi_\zeta$ is the map (8.9). Thus, the following equality holds:
$$
\begin{equation}
\begin{aligned} \, \notag &\operatorname{Prim}\mathfrak{T}/\mathrm{KL}_2(\mathbb{R})=\bigl\{\ker \pi (x,\psi)\colon x\in \mathbb{R},\, \psi=\pm\bigr\} \cup \bigl\{\ker \pi (\lambda)\colon \lambda \in \mathbb{R}\bigr\} \\ &\qquad \cup\bigl\{\ker \pi (\phi)\colon \phi=\pm \infty\bigr\} \cup \bigl\{\ker \pi (\phi,\psi,\zeta)\colon \phi,\,\psi= \pm \infty,\, \zeta \in b\mathbb{R}\bigr\}, \end{aligned}
\end{equation}
\tag{9.6}
$$
where the union is disjoint. For convenience of reference, we note that
$$
\begin{equation}
\begin{aligned} \, \notag &\operatorname{Prim} \mathfrak{T}/\mathcal{J}_\infty=\bigl\{\ker \pi (\lambda)\colon \lambda \in \mathbb{R}\bigr\} \cup\bigl\{\ker \pi (\phi)\colon \phi=\pm \infty\bigr\} \\ &\qquad \cup \bigl\{\ker \pi (\phi,\psi,\zeta)\colon \phi,\psi=\pm \infty,\, \zeta \in b\mathbb{R}\bigr\}. \end{aligned}
\end{equation}
\tag{9.7}
$$
Corollary 1. An operator $A \in \mathfrak{T}$ is Fredholm if and only if every operator of the form $\pi(A)$, where $\pi$ ranges over the representations (9.1) and (9.3)–(9.5), is invertible in the corresponding Hilbert space. Proof. By [7], Proposition 1.3.12, an operator $A \in \mathfrak{T}$ is Fredholm if and only if every operator $\rho(A)$ is invertible, where $\rho \in (\mathfrak{T}/\mathrm{KL}_2(\mathbb{R})) \,\widehat{}$ . The kernel of each representation $\rho \in (\mathfrak{T}/\mathrm{KL}_2(\mathbb{R})) \,\widehat{}$ coincides with one of the ideals (9.6). Therefore, the invertibility of the operator $\rho(A)$ for $\rho \in (\mathfrak{T}/\mathrm{KL}_2(\mathbb{R})) \,\widehat{}$ is equivalent to the invertibility of $\pi(A)$, where $\pi$ is a representation of one of the forms (9.1) and (9.3)–(9.5) such that its kernel coincides with $\ker \rho$.
The corollary is proved. § 10. The Jacobson topology on $\operatorname{Prim}\mathfrak{T}$It is convenient to represent the algebra $\mathfrak{T}$ as a restricted direct sum of two auxiliary $C^*$-algebras. Then the spectrum $\widehat{\mathfrak{T}}$ is obtained by gluing together the spectra of these algebras along some subspace. Let $\mathcal{A}_1, \mathcal{A}_2$ and $ \mathcal{C}$ be $C^*$-algebras and $\delta_{j}\colon\mathcal{A}_{j} \to \mathcal{C}$, $j=1,2$, be $*$-homomorphisms. The restricted direct sum of the algebras $\mathcal{A}_1$ and $\mathcal{A}_2$ is defined as the subalgebra
$$
\begin{equation*}
\mathcal{A}_1 \oplus_{\mathcal{C}}\mathcal{A}_2=\bigl\{(a_1,a_2) \in \mathcal{A}_1 \oplus \mathcal{A}_2\colon \delta_1(a_1)=\delta_2(a_2)\bigr\} \subset \mathcal{A}_1 \oplus \mathcal{A}_2.
\end{equation*}
\notag
$$
Let $X_1, X_2$ and $ Y$ be topological spaces and $i_{j}\colon Y \to X_j$, $j=1,2$, be continuous injective maps. Two elements $x_1$ and $x_2$ of the disjoint union $X_1 \sqcup X_2$ are said to be equivalent, $x_1\sim x_2$, if $x_1=x_2$ or there exists $y \in Y$ such that $x_1=i_1(y)$ and $x_2=i_2(y)$ or $x_1=i_2(y)$ and $x_2=i_1(y)$. The quotient space $X_1 \sqcup_Y X_2=X_1 \sqcup X_2 /{\sim}$ is called the gluing of the topological spaces $X_1$ and $X_2$ along $Y$. Suppose that the $*$-homomorphisms $\delta_{1}$ and $\delta_2$ are surjective. Then $\mathcal{C} \,{\cong}\, \mathcal{A}_{j} / \mkern-1mu\ker \delta_{j}$, and the spectrum of $\mathcal{C}$ is homeomorphic to a closed subset of the spectrum of each of $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$, that is, there are embeddings $\widehat{\delta_{j}}\colon \widehat{\mathcal{C}} \to \widehat{\mathcal{A}_{j}}$, $j=1,2$. According to [8], Theorem 2.16 (the formulation of this theorem contains a misprint: the condition of injectivity of the maps $\delta_j$ must be replaced by the condition of their surjectivity), there is a homeomorphism
$$
\begin{equation}
(\mathcal{A}_1 \oplus_{\mathcal{C}}\mathcal{A}_2) \,\widehat{} \, \cong \widehat{\mathcal{A}_1} \sqcup_{\widehat{\mathcal{C}}} \widehat{\mathcal{A}_2}.
\end{equation}
\tag{10.1}
$$
Let $\mathcal{J}^\infty$ be the ideal of $\mathfrak{T}$ generated by $F^{-1}C_0(\mathbb{R})F$, and let $\mathcal{J}_\infty$ be the ideal generated by $C_0(\mathbb{R})$. Set $\mathcal{A}_1=\mathfrak{T}/\mathcal{J}^\infty$, $\mathcal{A}_2=\mathfrak{T}/\mathcal{J}_\infty$ and $\mathcal{C}= \mathfrak{T}/(\mathcal{J}_\infty+ \mathcal{J}^\infty)$ and consider the quotient maps $\delta_j\colon \mathcal{A}_j \to \mathcal{C}$, $j=1,2$. From [27], Proposition 3.6, it follows that
$$
\begin{equation}
\mathfrak{T}/(\mathcal{J}_\infty \cap\mathcal{J}^\infty) \cong \mathfrak{T}/\mathcal{J}_\infty \oplus_{\mathcal{C}} \mathfrak{T}/\mathcal{J}^\infty.
\end{equation}
\tag{10.2}
$$
From this and equality (10.1) we obtain
$$
\begin{equation}
\operatorname{Prim}( \mathfrak{T}/(\mathcal{J}_\infty \cap\mathcal{J}^\infty)) \cong \operatorname{Prim} ( \mathfrak{T}/\mathcal{J}_\infty) \sqcup_{\operatorname{Prim} \mathcal{C}} \operatorname{Prim} ( \mathfrak{T}/\mathcal{J}^\infty).
\end{equation}
\tag{10.3}
$$
The spectrum $\operatorname{Prim} \mathfrak{T}/\mathcal{J}_\infty $ was considered in § 8. Lemma 10. There is an isomorphism between $\mathfrak{T}/\mathcal{J}^\infty$ and $\operatorname{SAP}(\mathbb{R})\oplus \operatorname{SAP}(\mathbb{R})$. The spectrum $\operatorname{Prim} \mathfrak{T}/\mathcal{J}^\infty $ consists of the ideals $\ker \pi (x,\psi)$, where $ x\in \mathbb{R}$ and $ \psi=\pm$, and $\ker \pi (\phi,\psi,\zeta)$, where $\phi,\psi=\pm \infty$ and $\zeta \in b\mathbb{R}$ (see (9.1) and (9.5)). Proof. Consider the map $\Phi^\infty\colon \mathfrak{T} \to \operatorname{SAP}(\mathbb{R}) \oplus \operatorname{SAP}(\mathbb{R})\colon aI+bS \mapsto a-b \oplus a+b$. Obviously, $\Phi^\infty$ is a surjective $*$-homomorphism, and thus it induces an embedding of $\operatorname{Prim}( \operatorname{SAP}(\mathbb{R})\oplus \operatorname{SAP}(\mathbb{R}))$ into $\operatorname{Prim}{\mathfrak{T}}$. The algebra $\operatorname{SAP}(\mathbb{R})\oplus \operatorname{SAP}(\mathbb{R})$ is commutative; therefore, for any of its irreducible representations $\rho$ the composition $\rho \circ \Phi^\infty$ coincides with one of the one-dimensional representations $\pi (x,\psi)$ or $\pi(\phi,\psi,\zeta)$. Conversely, it follows from (9.1) and (9.5) that each of $\pi (x,\psi)$ and $\pi(\phi,\psi,\zeta)$ is of the form $\rho \circ \Phi^\infty$, where $\rho$ is a one-dimensional representation of $\operatorname{SAP}(\mathbb{R})\oplus \operatorname{SAP}(\mathbb{R})$. Thus, the spectrum $\operatorname{Prim} \bigl( \operatorname{SAP}(\mathbb{R})\oplus \operatorname{SAP}(\mathbb{R}) \bigr)$ is in a one-to-one correspondence with the set
$$
\begin{equation}
\bigl\{\ker \pi (x,\psi)\colon x\in \mathbb{R},\, \psi=\pm\bigr\} \cup \bigl\{\ker \pi (\phi,\psi,\zeta)\colon \phi,\psi=\pm \infty,\, \zeta \in b\mathbb{R}\bigr\}.
\end{equation}
\tag{10.4}
$$
Clearly, each ideal in (10.4) contains $\mathcal{J}^\infty$. The complement to the set in (10.4) consists of the kernels of the representations $\pi(\lambda)$ and $\pi(\phi)$ from (9.3)-(9.4); it is open and corresponds to the spectrum of the ideal $\ker \Phi^\infty$. Note that none of the ideals $\ker \pi (\lambda)$ and $\ker \pi(\phi)$ contains $\mathcal{J}^\infty$, since the maps (9.3)-(9.4) do not vanish on $F^{-1}C_0(\mathbb{R})F \subset \mathcal{J}^\infty$. Consequently, the spectra of $\ker \Phi^\infty$ and $\mathcal{J}^\infty$ coincide, and therefore $\ker \Phi^\infty=\mathcal{J}^\infty$.
The lemma is proved. Lemma 11. The spectrum of $\operatorname{SAP}(\mathbb{R})$ consists of the maps Proof. The set of characters of $\operatorname{SAP}(\mathbb{R})$ was essentially described in the proof of Lemma 10. We need only to describe the topology on the spectrum. Consider the $*$-homomorphism $\beta \colon C(\overline{\mathbb{R}})\otimes \operatorname{AP}(\mathbb{R})\to \operatorname{SAP}(\mathbb{R})$ defined on generators by Clearly, $\beta$ is a surjection and therefore induces an embedding of $\widehat{\operatorname{SAP}(\mathbb{R})}$ into the spectrum of $C(\overline{\mathbb{R}})\otimes \operatorname{AP}(\mathbb{R})$. The Gelfand isomorphism where $\gamma(f)(\zeta)=\pi_\zeta(f)$, leads to an isomorphism $C(\overline{\mathbb{R}})\otimes \operatorname{AP}(\mathbb{R}) \cong C(\overline{\mathbb{R}})\otimes C(b\mathbb{R})$. The map $f\otimes g \mapsto h$, where $h(x,\zeta)=f(x)g(\zeta)$, induces an isomorphism $C(\overline{\mathbb{R}})\otimes C(b\mathbb{R}) \cong C(\overline{\mathbb{R}}\times b\mathbb{R})$. Therefore, the spectrum of $(C(\overline{\mathbb{R}})\otimes \operatorname{AP}(\mathbb{R}))\,\widehat{}$ is homeomorphic to $\overline{\mathbb{R}}\times b\mathbb{R}$. The homeomorphism is implemented by where $\rho(x,\zeta)$ acts on the generators by the rule $f\otimes e_k \mapsto f(x)\cdot \pi_\zeta(e_k)$.
Let us check that the embedding (10.7)-(10.8) coincides with the restriction to $\widehat{\operatorname{SAP}(\mathbb{R})}$ of the inverse of the homeomorphism (10.9). Setting $x\in \mathbb{R}$ and $\zeta=\eta_x$ in (10.9) we obtain $\rho(x,\eta_x)=\tau(x)\circ \beta$. Similarly, for $x=\pm \infty$ and arbitrary $\zeta$ we have $\rho(\pm \infty,\zeta)=\tau(\pm \infty,\zeta) \circ \beta$. The lemma is proved. Let $b\mathbb{R}^\pm$ be copies of $b\mathbb{R}$. Consider the disjoint union $\check{\Omega}_0=b\mathbb{R}^{-}\cup \mathbb{R} \cup b\mathbb{R}^{+}$, which parametrizes the spectrum $\widehat{\operatorname{SAP}(\mathbb{R})}$ (see (10.5)-(10.6)). Put $\check{\Omega}=\check{\Omega}_+\sqcup \check{\Omega}_-$ with the topology of the disjoint union, where $\check{\Omega}_\pm$ are copies of $\check{\Omega}_0$. Corollary 2. The spaces $\operatorname{Prim} \mathfrak{T}/\mathcal{J}^\infty$ and $\check{\Omega }$ are homeomorphic. The homeomorphism is implemented by the map Lemma 12. The algebra $\mathcal{C}$ is isomorphic to $\mathbb{C}^4 \otimes C(b\mathbb{R})$. The spectrum $\operatorname{Prim} \mathcal{C}$ consists of the ideals $\ker \pi (\phi,\psi,\zeta)$ for $\phi,\psi=\pm \infty$ and $ \zeta \in b\mathbb{R}$. Proof. The spectrum $\operatorname{Prim} \mathcal{C}$ is the intersection of $\operatorname{Prim} \mathfrak{T}/\mathcal{J}^\infty$ and $\operatorname{Prim} \mathfrak{T}/\mathcal{J}^\infty$, and thus it coincides with the space $\bigl\{\ker \pi (\phi,\psi,\zeta)\colon \phi,\psi=\pm \infty,\zeta \in b\mathbb{R}\bigr\}$, which is homeomorphic to the disjoint union of four copies of $b\mathbb{R}$. The map is an isomorphism of the commutative algebras $\mathcal{C}$ and $\mathbb{C}^4 \otimes C(b\mathbb{R})$.
The lemma is proved. Proof. We have $\operatorname{Prim}\mathfrak{T}/(\mathcal{J}_\infty \cap\mathcal{J}^\infty)= \operatorname{Prim}\mathfrak{T}/\mathcal{J}^\infty \cup \operatorname{Prim}\mathfrak{T}/\mathcal{J}_\infty$. Lemma 10 and equalities (9.6)-(9.7) lead to the equality $ \operatorname{Prim}\mathfrak{T}/\mathcal{J}^\infty \cup \operatorname{Prim}\mathfrak{T}/\mathcal{J}_\infty=\operatorname{Prim} \mathfrak{T}/\mathrm{KL}_2$. Thus, $\mathcal{J}_\infty \cap\mathcal{J}^\infty=\mathrm{KL}_2$.
The lemma is proved. Let $b\mathbb{R}^\pm_\pm$ be copies of $b\mathbb{R}$. Using the isomorphism (10.13) we identify the spectrum $\operatorname{Prim}\mathcal{C}$ with the disjoint union $b\mathbb{R}^+_+\sqcup b\mathbb{R}^+_-\sqcup b\mathbb{R}^-_+\sqcup b\mathbb{R}^-_-$. Set
$$
\begin{equation}
i_{1}\colon b\mathbb{R}_\pm^\pm \ni \eta \mapsto \eta \in b\mathbb{R}_\pm^\pm \in \check{\Omega},
\end{equation}
\tag{10.14}
$$
$$
\begin{equation}
i_{2}\colon b\mathbb{R}_+^\pm \ni \eta \mapsto (Q_3(\pm\infty,+\infty),\eta) \in \Omega
\end{equation}
\tag{10.15}
$$
and
$$
\begin{equation}
i_{2}\colon b\mathbb{R}_-^\pm \ni \eta \mapsto (Q_3(\pm\infty, -\infty),\eta) \in \Omega.
\end{equation}
\tag{10.16}
$$
Summarizing, we obtain the following statement. 
Citation in format AMSBIB
\Bibitem{BaiSar24} |
|
||||||||||||||||||||||||||||
|
Contact us: |
Terms of Use
|
Registration to the website |
Logotypes |
|