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Automatic continuity of a locally bounded homomorphism of Lie groups on the commutator subgroup A. I. Shternabc a Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia b Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia c Scientific Research Institute for System Studies of the Russian Academy of Sciences, Moscow, Russia
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     § 1. IntroductionThis paper is dedicated to the blessed memory of O. V. Smyka. One of the well-known results associated with the phenomena of automatic continuity states that the restriction of a locally bounded linear representation of a connected Lie group to the commutator subgroup of this group is continuous (see [1]–[3]). In this paper, this statement is extended to arbitrary locally bounded homomorphisms of Lie groups. The main technical tool is the concept of the discontinuity group of a homomorphism of topological groups, which is recalled in the next section. § 2. Preliminaries2.1. The discontinuity group of a locally relatively compact homomorphism of topological groupsFirst let us recall the definitions of the local relative compactness and local boundedness of a homomorphism of topological groups. Definition 1. Let $G$ be a topological group, and let $\pi$ be a (not necessarily continuous) homomorphism of $G$ into a Hausdorff topological group $H$. We say that $\pi$ is locally relatively compact if there exists a neighbourhood $V$ of the identity element $e$ of $G$ such that the closure of the set $\pi(V)$ is a compact subset of the topological group $H$. We say that $\pi$ is locally bounded if there exists a neighbourhood $V$ of the identity element $e$ of $G$ such that $\pi(V)$ is a completely bounded subset of the topological group $H$, that is, for any neighbourhood $W$ of the identity element $e_H$ there is a finite number of shifts of $W$ in $H$ whose union covers $\pi(V)$. Remark 1. Clearly, if the group $H$ is locally compact, then any locally bounded homomorphism into $H$ is automatically locally relatively compact. Definition 2. Let $\mathfrak U=\mathfrak U_G$ be the descending (with respect to inclusion) family of neighbourhoods of the identity in a Hausdorff topological group $G$. For every locally relatively compact (but not necessarily continuous) homomorphism $\pi$ of $G$ into a Hausdorff topological group $H$ we introduce the notation Here and below a bar means the closure with respect to the corresponding topology (in this case, in the topology of the group $H$). Theorem 1. Let $G$ be a topological group, and let $\pi$ be a locally relatively compact homomorphism of $G$ into a Hausdorff topological group $H$. Then the set $\operatorname{DG}(\pi)$ is a compact subgroup of the topological group $H$ and a compact normal subgroup in the closed subgroup $\overline{\pi(G)}$ of $H$. Moreover, for any neighbourhood $V$ of $\operatorname{DG}(\pi)$ there is a neighbourhood $U$ of the identity element of $H$ such that $\overline{\pi(U)}\subset V$, and the homomorphism $\pi$ is continuous if and only if $\operatorname{DG}(\pi)=\{e_H\}$. Proof. The first statement was proved in [3], Theorem 1.1.2.
Let $V$ be a neighbourhood of the set $\operatorname{DG}(\pi)$ in $H$, and let $U_0$ be a neighbourhood of the identity element of $G$ for which, by assumption, the set $\overline{\pi(U_0)}$ is compact in $H$. Hence the set $\overline{\pi(U_0)}\setminus V$ is also compact in $H$. The family of intersections $(\overline{\pi(U_0)}\setminus V)\cap \overline{\pi(U)}$, $U\in\mathfrak U$, has an empty intersection and, by the compactness of the set $\overline{\pi(U_0)}\setminus V$, contains a finite number of members $(\overline{\pi(U_0)}\setminus V)\cap \overline{\pi(U)}$ with an empty intersection. We denote the corresponding neighbourhoods by $U_1,\dots,U_n$. Then the intersection of the sets $(\overline{\pi(U_0)}\setminus V)\cap \overline{\pi(U)}$ with $U \in\mathfrak U$ such that $U\subset\bigcap_{i\colon 1\leqslant i\leqslant n}U_i$ is empty, so that $\overline{\pi(U)}\subset V$. The fact that the condition of the continuity of the homomorphism $\pi$ is equivalent to the condition $\operatorname{DG}(\pi)=\{e_H\}$ was proved in [3], Theorem 1.1.2. This completes the proof of the theorem. Proposition 1. Let $\mathfrak U=\mathfrak U_G$ be the descending (with respect to inclusion) family of neighbourhoods of the identity in a topological group $G$. For any locally relatively compact (but not necessarily continuous) homomorphism $\pi$ of $G$ into a Hausdorff topological group $H$ consider the net $\{\pi(g_U)\mid g_U\in U\in\mathfrak U_G\}$ in $H$ (for nets and their convergence, see [4], § 1.6, and Proposition 1.6.1). Then all limit points of the net $\{\pi(g_U)\}$ belong to $\operatorname{DG}(\pi)$. Proof. By Theorem 1, for any neighbourhood $V$ of the set $\operatorname{DG}(\pi)$ introduced above there is a neighbourhood $U$ of the identity of $G$ such that $\overline{\pi(U)}\subset V$. As is known, the Hausdorff space of a topological group is completely regular, so that every closed subset $A$ of $H$ has a neighbourhood that does not contain a given point $x_0$ not belonging to $A$. Therefore, the intersection of all neighbourhoods of $\operatorname{DG}(\pi)$ coincides with $\operatorname{DG}(\pi)$. In turn, if $V$ is a neighbourhood of $\operatorname{DG}(\pi)$ in $H$ and $U_0$ is a neighbourhood of the identity of $G$ such that $\overline{\pi(U_0)}\subset V$, then for $U\subset U_0$ we have a fortiori $\overline{\pi(U)}\subset V$, so that all the limit points of the net $\{\pi(g_U)\}$ lie in the closure of $V$ for any neighbourhood $V$ of the set $\operatorname{DG}(\pi)$, and therefore all of them lie in $\operatorname{DG}(\pi)$.
This completes the proof of the proposition. Proposition 2. Let $\mathfrak U=\mathfrak U_G$ be the descending (with respect to inclusion) family of neighbourhoods of the identity of a topological group $G$. Let $\pi$ be a locally relatively compact (but not necessarily continuous) homomorphism of $G$ into a Hausdorff topological group $H$. Then every point of $\operatorname{DG}(\pi)$ is the limit of some subnet of the net $\{\pi(g_U)\}$. Proof. Let $V\in H$ be a neighbourhood of the set $\operatorname{DG}(\pi)$ in $H$, and let $U\in\mathfrak U_G$ and $W\in\mathfrak U_H$. Let $h_0\in\operatorname{DG}(\pi)$. We assign an element $h_{(V,U, W)}\in h_0W\cap{\overline{\pi(U)}}$ to the triple $(V, U, W)$ (the set $h_0W\cap{\overline{\pi(U)}}$ is nonempty since $h_0\in\overline{\pi(U)}$). According to [4], Theorem 3.1.23, the net thus constructed (with nets in every component in descending order with respect to inclusion) has a limit point, and the limit of the corresponding subnet certainly coincides with $h_0$ in view of the participation of a net with $W\in\mathfrak U_H$.
This completes the proof of the proposition. Definition 3. Let $G$ be a topological group, and let $\pi$ be a locally relatively compact homomorphism of $G$ into a topological group $H$. Then the compact normal subgroup $\operatorname{DG}(\pi)$ of the closure of the $\pi$-image of the group $G$ is called the discontinuity group of the homomorphism $\pi$. In particular, the discontinuity group is defined for any locally bounded homomorphism into a locally compact group (see Remark 1), and thus also for any locally bounded finite-dimensional representation of a topological group. Definition 4. Let $G$ be a group and $X$ be a subset of $G$. The set $X$ is said to be infinitely divisible if for any element $x\in X$ and every positive integer $p$ there is an element $y\in X$ such that $y^p=x$. A group $G$ is called locally infinitely divisible if the operation of raising to any positive integer power $p$ is open at the identity element of the group, that is, the set of $p$th powers of elements of any neighbourhood of identity in $G$ contains some neighbourhood of identity in $G$. Every Lie group is obviously locally infinitely divisible. The following folklore statement shows that in every topological group the closures of relatively compact infinitely divisible subgroups are infinitely divisible. Lemma 1. Let $G$ be a topological group and $X$ be a subset of $G$ whose closure in $G$ is compact. If $X$ is infinitely divisible, then so is its closure in $G$. This statement was proved, for example, in [3], Lemma 1.1.5. Lemma 2. Let $G$ be a locally infinitely divisible group and $\pi$ be a locally relatively compact homomorphism of $G$ into a Hausdorff topological group $H$. Then the discontinuity group $\operatorname{DG}(\pi)$ is a compact connected subgroup of $H$. This was proved in [3], Lemma 1.1.6. The proofs of statements below uses the following factorization property of discontinuity groups. Lemma 3. Let $G$ be a connected locally compact group, $N$ be a closed normal subgroup of $G$, and let $\pi$ be a locally relatively compact homomorphism of $G$ into a Hausdorff topological group $H$ (for example, a locally bounded homomorphism into a locally compact group; see Remark 1). Let $\operatorname{DG}(\pi|_N)$ be the discontinuity group of the restriction $\pi|_N$. Then $\operatorname{DG}(\pi|_N)$ is a closed normal subgroup of the compact discontinuity group $\operatorname{DG}(\pi)$, and the corresponding quotient group $\operatorname{DG}(\pi)/\operatorname{DG}(\pi|_N)$ is isomorphic to the discontinuity group $\operatorname{DG}(\psi)$ of the homomorphism $\psi$ of the group $G$ obtained by the composition of $\pi$ and the canonical homomorphism This was proved in [3], Lemma 1.1.7. The following property connects the discontinuity group of a homomorphism and that of the homomorphism of the commutator subgroup defined by this homomorphism. Lemma 4. Let $G$ be a topological group, $G'$ be the commutator subgroup of $G$ (in the topology induced by the group topology), and let $\pi$ be a locally relatively compact homomorphism of $G$ into a Hausdorff topological group $H$. Then the commutator subgroup of the discontinuity group of $\pi$ is contained in the discontinuity group of the restriction $\pi|_{G'}$ of $\pi$ to the commutator subgroup $G'$ of $G$: If the discontinuity group $\operatorname{DG}(\pi)$ is commutative, then the discontinuity group of the restriction $\pi|_{G'}$ of $\pi$ to $G'$ is the identity group, that is, This statement was proved in [3], Lemma 1.1.9. The following statement enables us to describe the structure of the discontinuity group of finitely decomposable topological groups. Lemma 5. Let $G$ be a topological group, let $A_1,\dots,A_n$ be closed subgroups of $G$, and let $A=A_1\times A_2\times\cdots\times A_n$. Suppose that the map $A\to G$ defined by the rule $(a_1,a_2,\dots,a_n)\mapsto a_1a_2\cdots a_n\in G$ for any $(a_1,a_2,\dots,a_n)\in A$, $a_i\in A_i$, $i=1,\dots,n$, is open at the identity element of the group $A$. Let $\pi$ be a locally relatively compact homomorphism of $G$ into a Hausdorff topological group $H$. Then any element $d$ of the discontinuity group $\operatorname{DG}(\pi)$ can be represented in the form $d=d_1d_2\cdots d_n$, where $d_i\in\operatorname{DG}(\pi|_{A_i})$. This statement was proved in [3], Lemma 1.1.12. 2.2. Continuity conditions for some homomorphisms of topological groupsTheorem 2. Let $G$ and $H$ be topological groups, and let $f$ be a locally relatively compact homomorphism of $G$ into the Hausdorff group $H$. Let $M$ and $N$ be closed normal subgroups of $H$ such that the intersection $M\cap N$ does not contain nontrivial compact subgroups, and let $\varphi$ and $\psi$ be the canonical homomorphisms of the group $H$ onto the quotient groups $H/M$ and $H/N$, respectively. If the compositions $\varphi\circ f$ and $\psi\circ f$ are continuous, then so is the homomorphism $f$. This was proved in [3], Theorem 1.1.13. § 3. An auxiliary resultTheorem 3. The discontinuity group of every locally bounded homomorphism of a Lie group into a Lie group is commutative. This was proved in [5], Theorem 2, using the statements above and the properties of Lie algebras of Lie groups (see [6]). Remark 2. Let $G$ and $H$ be Lie groups, and let $\pi\colon G\to H$ be a homomorphism. Since the identity component $G_0$ is open in $G$ and $H_0$ is open in $H$, it follows that in investigations of the continuity of $\pi$ we can assume that $G$ and $H$ are connected. The proof in [5], Theorem 2, lacks an explanation of the fact that the discontinuity group of the homomorphism under consideration is contained in the full preimage of the discontinuity groups $DG(\rho)$ of the adjoint representation $\rho$ of the group $H$ (see [6], Ch. IX, § 3.5) under the canonical homomorphism of the group $H$ onto its quotient group by its centre $Z_H$. This follows immediately from Lemma 3 as applied to the centre $Z_H$ of the group $H$. § 4. Main theoremTheorem 4. Every locally bounded homomorphism of a Lie group $G$ into a Lie group is continuous on the commutator subgroup $G'$ of $G$. Proof. Let $\pi$ be a locally bounded homomorphism of the Lie group $G$ into a Lie group $H$. Then $\pi$ is locally relatively compact (see Remark 1). As proved in Theorem 3, the discontinuity group $\operatorname{DG}(\pi)$ of $\pi$ is commutative. By Lemma 4
$$
\begin{equation*}
\operatorname{DG}(\pi|_{G'})=\operatorname{DG}(\pi)'=\{e_G\},
\end{equation*}
\notag
$$
and the homomorphism $\pi$ is continuous on $G'$ by Theorem 1.
This completes the proof of the theorem. § 5. Some applicationsRemark 3. Every locally bounded homomorphism of a Lie group into a Lie group with no nontrivial connected compact subgroups is continuous. Proof. By Lemma 2 the discontinuity group $\operatorname{DG}(\pi)$ of such a homomorphism $\pi$ is connected and compact. Therefore, it is trivial, and $\pi$ is continuous by Theorem 1. Theorem 5. A locally bounded homomorphism of a connected Lie group $G$ whose commutator subgroup $G'$ is closed and admits a closed additional subgroup $Z$ such that $G=G'Z$, into a connected Lie group is continuous (in the intrinsic Lie topology) if and only if its restriction to $Z$ is continuous. Proof. If a homomorphism of a group is continuous, then it is continuous on every subgroup of this group, and thus it is sufficient to prove the ‘if’ part.
Let $G$ be a connected Lie group whose commutator subgroup $G'$ admits a closed additional subgroup $Z$ such that $G=G'Z$. Let $H$ be a connected Lie group. Let $\pi$ be a locally bounded homomorphism of $G$ into $H$ which is continuous on $Z$. The discontinuity group of the restriction of $\pi$ to the commutator subgroup $G'$ is the identity group (see Theorem 3), and therefore, by Theorem 4, the restriction of $\pi$ to the commutator subgroup $G'$ is continuous with respect to the intrinsic Lie topology (cf. Theorem 1.1.2 in [3] and the correction in [1].) Consequently, $\pi$ is separately continuous with respect to the subgroups $Z$ and $G'$. By Namioka’s theorem (see [7]) $\pi$ has a point of joint continuity and therefore is continuous. This completes the proof of Theorem 5. Corollary 1. Let $G$ be a connected Lie group that is either linear or simply connected, and let its commutator subgroup $G'$ admit a closed additional subgroup $Z$ such that $G=G'Z$. A locally bounded homomorphism of $G$ into a Lie group is continuous if and only if it is continuous on $Z$. Proof. If a homomorphism of a group is continuous, then it is obviously continuous on each subgroup of this group, and thus it is sufficient to prove the ‘if’ part.
Let $G$ be a connected Lie group that is either linear or simply connected, and let the commutator subgroup $G'$ of $G$ admit a closed additional subgroup $Z$ such that $G=G'Z$. Then the commutator subgroup $G'$ is closed in $G$ (see [8], Theorem 3.8.12 and Exercise 41, (e), to Ch. 3). Now the statement of the corollary follows from Theorem 5. This completes the proof. Corollary 2. Let $G$ be a semidirect product of a perfect Lie group $B$ and a commutative Lie group $Z$. Then a locally bounded homomorphism of $G$ into a Lie group is continuous if and only if it is continuous on $Z$. Proof. If a homomorphism of a group is continuous, then it is obviously continuous on every subgroup of the group, and thus it is sufficient to prove the ‘if’ part.
Let $B$ be a perfect Lie group, $Z$ be an Abelian Lie group, and let a connected Lie group $G$ be included in the split short exact sequence
$$
\begin{equation*}
\{e\}\to Z\overset{\iota}\to G\overset{\rho}\to B\to\{e\},
\end{equation*}
\notag
$$
where $\iota$ is a continuous embedding and $\rho$ is the canonical epimorphism of $G$ onto the group $B$, which is isomorphic to the quotient group $G/Z$. Then the commutator subgroup $G'$ of $G$ is mapped by $\rho$ onto the commutator subgroup $B'$ of $B$. Moreover, $G'$ is in a natural one-to-one correspondence with $B'$. Indeed, for any ${z_1,z_2\in Z}$ and $b,c\in G$ we have $bz_1c_2(bz_1)^{-1}(cz_2)^{-1}=bzb^{-1}c^{-1} =[b,c]$, and therefore for any $b,c\in Z$ the commutator of $bZ$ and $cZ$ is equal to $[b,c]Z$. Thus, the commutator subgroup of the group $G$ is naturally isomorphic to the commutator subgroup of $B$. However, the group $B$ is perfect, which means that $B'=B$. Therefore, the subgroup $G'$ of $G$ coincides with the isomorphic image of $B$ in $G$ under the splitting map and is therefore closed, and every element of $G$ is the product of an element of $G'$ and an element of $Z$.
Now the statement of the corollary follows from Theorem 5. This completes the proof of the corollary. Theorem 6. Let $G$ be a connected Lie group, $G'$ be the commutator subgroup of $G$, $\mathfrak g$ be the Lie algebra of $G$, $\mathfrak g'$ be the commutator subalgebra of $\mathfrak g$, $\mathfrak h$ be a vector subspace of $\mathfrak g$ complementary to $\mathfrak g'$, $\{h_1,\dots,h_k\}$ be a basis in $\mathfrak h$, and let the one-dimensional vector subspaces $\mathfrak h_1,$ $\dots,$ $\mathfrak h_k$ be spanned by $h_1,\dots,h_k$, respectively. Let $G'$ be closed in $G$. Then a locally bounded homomorphism $\pi$ of $G$ into a Lie group $G_1$ is continuous if and only if the composite maps of $\mathbb R$ into $G_1$ given by the formulae $\mathbb R\ni t\mapsto \pi(\exp(th_i))$, $i=1,\dots, k$, are continuous. Proof. If $\pi$ is continuous, then each composite map from $\mathbb R$ into $G_1$ given by the formula $\mathbb R\ni t\mapsto \pi(\exp(th_i))$, $i=1,\dots, k$, is continuous. This proves the ‘only if’ part of the theorem. Therefore, it remains to prove the converse statement.
We prove the ‘if’ part. Let the corresponding conditions of the theorem be satisfied for some vector subspace $\mathfrak h$ in $\mathfrak g$ complementary to $\mathfrak g'$ and for some basis $\{h_1,\dots,h_k\}$ of $\mathfrak h$. By Theorem 4 the restriction of the representation $\pi$ to $G'$ is continuous (since $G'$ is closed, the intrinsic topology of the Lie group $G'$ coincides with the topology induced from $G$). On the other hand, by the assumptions of the ‘if’ part of the theorem the composite maps of $\mathbb R$ into $G_1$ given by $\mathbb R\ni t\mapsto \pi(\exp(th_i))$, $i=1,\dots, k$, are continuous. Consequently, the homomorphism $\pi$ is separately continuous in a neighbourhood of the identity element $e$ of the group $G$ with respect to the subgroups $G'$ and $H_1,\dots,H_k$. Since the corresponding map defined by the restriction of the product of the corresponding exponential maps of subgroups to a small neighbourhood $U$ of the identity element $e$ of $G$ is an analytic diffeomorphism of $(G'\times H_1\times\dots\times H_k)\cap U$ onto a neighbourhood in $G$ by Theorem 2.10.1 of [8], it follows from Namioka’s theorem (see [7]) that the representation $\pi$ has a point of joint continuity at $U$ and is therefore continuous. This completes the proof of Theorem 6. AcknowledgementThe author is deeply grateful to an anonymous referee for pointing out some inaccuracies and giving numerous tips to improve the presentation. 
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