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A criterion for the weak continuity of representations of topological groups in dual Fréchet spaces A. I. Shternabc a Lomonosov Moscow State University, Faculty of Mechanics and Mathematics b Moscow Center for Fundamental and Applied Mathematics c Scientific Research Institute for System Studies of RAS, Moscow
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    § 1. IntroductionBeginning with Banach’s theorem of 1932 (Theorem 1.4 in [1]) stating that a Baire measurable homomorphism of one complete separable metrizable group into another is continuous, conditions for continuity of measurable, Borel and Baire homomorphisms and representations of topological groups and semigroups were studied (see, for example, [2]–[10], related results in [11], and the erroneous paper [12]). However, the measurability condition for a mapping is restrictive, and the verification of this condition is not always harmless. For example, if $K$ is a locally compact space equipped with a continuous action $\alpha$ of an amenable (see, for example, [12]–[14]) locally compact group $G$, $f$ is a continuous function on $K$, and the function $F$ is defined as the result of applying some invariant mean on $G$ to a function of two variables on $G\times K$ given by the formula $(g,k)\mapsto f(\alpha(g)k)$, $g\in G$, $k\in K$, then the question of the measurability of this function $F$ on $K$ is not resolved even for the case in which $K$ is compact and $G$ is the group of integers; in any case, even for a compact space $K$, one cannot guarantee the Borel measurability (a corresponding example is given in Remark 1 in [15] and, in a somewhat simplified form, in the review [17] of the paper [16]), and, all the more, it is unclear how to ensure the continuity of the function $F$. Methods of Banach algebra theory are sometimes quite useful in solving various measurability problems. However, the complexity of the situation increases when considering not necessarily bounded representations of noncommutative groups; in this case, from the point of view of the Banach algebra theory, it is necessary to use less studied weighted group algebras (on general locally compact groups), for which, currently, some important tools are missing, including the criterion for their amenability. Therefore, it is advisable, on the one hand, to restrict ourselves to the means of the group language itself and, on the other hand, to refuse the strong assumption of measurability. In this connection, in the present paper, as in [18], where we have considered representations of topological groups in Banach spaces, the continuity conditions are studied on the basis of estimating the value of the weak$^*$ variation at a point. It is tacitly assumed that the group under consideration is a Hausdorff topological group. In particular, we prove the following criterion for continuity of representations of topological groups by bounded linear operators in a Fréchet space dual to some locally convex space. Theorem 1. Let $G$ be a topological group and let $\pi$ be a (group-theoretic) representation of $G$ in a Fréchet space $E$ dual to some locally convex space $E_*$ (in the strong topology) by operators adjoint to continuous linear operators on $E_*$ (which will be briefly called adjoint operators). Then the representation $\pi$ is continuous in the weak$^*$ operator topology1 if and only if, for some $q$, $0\leqslant q<1$, there is a neighbourhood $V$ of the neutral element $e$ of $G$ such that, for any neighbourhood $U$ of the zero element in $E$ and its polar $\mathring{U}$ in $E^*$ and, for any vector $\xi$ in $U$ and any element $\varphi\in\mathring{U}$, the inequality $|(\pi(g)\xi- \xi)(\varphi)|\leqslant q$ holds for each $g\in V$. Thus, if the weak variation of any representation by adjoint operators of any topological group on a Fréchet space dual to some locally convex space satisfies the inequality given in the theorem, then this variation is zero, and the representation is continuous. A natural question arises about the possibility of improving the value of the constant $q$, which secures weak continuity of the representation. An obvious example of the unitary representation of the group $\mathbb R$ by shifts in the space $l^2(\mathbb{R})$ shows that it is impossible to replace the condition $q<1$ by the condition $q\leqslant\sqrt2$. As is known [18], the representations of locally compact groups in finite-dimensional normed spaces admit an improvement in the general estimate for $q$. In the next section, we obtain several auxiliary statements and prove the main theorem. Some results of the paper were announced in [19]. § 2. Notation, auxiliary statements, and the proof of the main theoremLet $\mathbb R$ be the field of real numbers and let $\mathbb N$ be the semigroup of positive integers. The symbol $\mathbb R$ is also used to denote the additive group of the corresponding field. Definition 1. Let $G$ be a topological group and let $\pi$ be a representation (not necessarily continuous in any operator topology) of $G$ in a locally convex space $E$ which is dual to a locally convex space $E_*$. According to the bipolar theorem (see, for example, [20], Theorem, IV.1.5 and Corollary 3 to this theorem), the topology of $E$ has a neighbourhood base formed by sets coinciding with their $(E,E_*)$-bipolars. The weak$^*$ variation $\omega^*(\pi;\xi;f;U)\geqslant0$ of the representation $\pi$ in a neighbourhood $V$ of the identity element $e$ of the group $G$ on a vector $\xi\in E$ and a functional $\varphi\in E_*$ is defined by $\sup_{g\in V}|(\pi(g)\xi-\xi)(\varphi)|$; the weak$^*$ variation $\omega^*(\pi;\xi;\varphi)\geqslant0$ of the representation $\pi$ at the identity element $e$ of the group $G$ on a vector $\xi\in E$ and a functional $\varphi\in E_*$ is defined as the infimum of $\sup_{g\in V}|(\pi(g)\xi-\xi)(\varphi)|$ over all neighbourhoods $V\subset G$ of the identity element $e\in G$, that is, the weak$^*$ variation $\omega^*(\pi)\geqslant0$ of the representation $\pi$ at the identity element $e$ of the group $G$ (or, briefly, the weak$^*$ variation of the representation $\pi$) is defined as the supremum over all neighbourhoods $U$ of the zero element in $E$ and over all vectors $\xi\in U$ and all functionals $\varphi\in E_*$ from the polar $\mathring{U}$ of the neighbourhood $U$ of the lower bounds of the quantities $\sup_{g\in V}|(\pi(g)\xi-\xi)(\varphi)|$ over all neighbourhoods $V\subset G$ of the identity element $e\in G$,Definition 2. Let $G$ be a topological group and let $\pi$ be a (group–theoretic) representation of $G$ in a Fréchet space $E$. We say that $\pi$ is locally uniformly bounded if there is a neighbourhood $V_0$ of the identity element $e$ in the group $G$ such that the restriction of $\pi$ to $V_0$ is an equicontinuous family of continuous linear operators in $E$ (cf. [21], Proposition 1, and [22], pp. 97–98). We need the following topology in the space of continuous linear operators in the representation space. Definition 3. Let $E$ be a Fréchet space dual to some locally convex space $E_*$ and let $\mathcal {L}(E)$ be the space of continuous linear operators on $E$. By the weak$^*$ operator topology in $\mathcal {L}(E)$ we mean the topology defined by the family of seminorms of the form $T\mapsto|(T\xi)(\varphi)|$, where $T\in\mathcal {L}(E)$, $\xi\in E$, and $\varphi\in E_*$. In general, the space $E_*$ is not unambiguously determined by the space $E$, and so the weak$^*$ operator topology also depend on the choice of $E_*$. Thus, the above actual topology depends on the choice of the predual space of $E$ if it is not unique up to an isomorphism. Lemma 1. Let $G$ be a topological group and let $\pi$ be a locally uniformly bounded representation of $G$ in a locally convex space $E$. Let $E$ be dual to some locally convex space $E_*$, let the representation operators of $\pi$ be adjoint to some continuous linear operators in the predual space $E_*$, and let $\omega^*(\pi)$ be the weak$^*$ variation of the representation $\pi$ introduced in Definition 1. 1. If $a\in\mathbb{R}$, $a>0$, then $\omega^*(\pi)\leqslant a$ if and only if, for any $b>a$ and any neighbourhood of zero $U$ in $E$, there is a neighbourhood $V\subset G$ of the identity element $e\in G$ such that $|f(\pi(g)\xi-\xi)|< b$ for each $f$ in the polar $\mathring{U}$ of the neighbourhood $U$, each vector $\xi\in U$ in the space $E$ of the representation $\pi$, and each $g\in V$. 2. The representation $\pi$ is continuous in the weak$^*$ operator topology if and only if $\omega^*(\pi)=0$. 3. If the value $\omega^*(\pi)$ is finite, then the representation $\pi$ is locally uniformly bounded. In particular, if the group $G$ is compact and $\omega^*(\pi)$ is finite, then the representation $\pi$ is bounded. Proof. Assertion 1 is immediate from the definition of $\omega^*(\pi)$ (see formula (1)), and assertion 2 from the definition of the weak$^*$ operator topology (see Definition 3 below) and from assertion 1.
Let us prove assertion 3. Assume that the supremum over all neighbourhoods $U$ of the zero element in $E$ and all vectors $\xi\in U$ and all functionals $\varphi\in E_*$ from the polar $\mathring{U}$ of the neighbourhood $U$, of the lower bounds of the quantities $\sup_{g\in V}|(\pi(g)\xi-\xi)(\varphi)|$ over the neighbourhoods $V\subset G$ of the identity element $e\in G$ is finite. This implies that the set $\{\pi(g),\,g\in V\}$ of the vector $\xi$ is absorbed by a multiple of the bipolar of the neighbourhood $U$ and thus bounded, since $U$ is arbitrary. Thus, the set $\{\pi(g),\, g\in U\}$ is bounded in the weak$^*$ operator topology and, therefore, is equicontinuous (see Theorem 4.2 in [20]). The remark about the compact group is obvious. This completes the proof of Lemma 1. Lemma 2. Let $G$ be a locally compact group, let $H_i$, $i=1,\ldots,n$, be a finite family of metrizable locally compact groups, and let $\psi_i\colon H_i\to G$ be homomorphisms such that the restriction $\psi_V$ of the mapping $\psi$ of the product $H=H_1\times \dots \times H_n$ to $G$ defined by $\psi(h_1,\dots,h_n) =\psi_1(h_1) \cdots \psi_n(h_n)$ (where $h_i\in H_i$ for all $i=1,\dots,n$) to some open neighbourhood $V$ of the form $V=\prod_{i=1}^n V_i$ of the identity element $e_H$ of $H=\prod_{i=1}^n H_i$, where $V_i$ are open neighbourhoods of the identity elements $e_i$ of the groups $H_i$, $i=1, \dots,n$, is a homeomorphism onto some open neighbourhood of the identity element in $G$. Let $\pi$ be a (group-theoretic) representation by adjoint operators of the group $G$ in a Fréchet space dual to a locally convex space. If the representation $\pi\circ\psi_i$ is continuous in the weak$^*$ operator topology on the neighbourhood $V_i$ of the identity element $e_i$ in $H_i$ for each of the groups $H_i$, $i=1, \dots,n$, then the representation $\pi$ is continuous in the weak$^*$ operator topology. In particular, if $G$ is a connected Lie group and the representation $\pi$ of $G$ is continuous in the weak$^*$ operator topology on every one-parameter subgroup of the group $G$ in the intrinsic Lie topology of this analytic subgroup, then $\pi$ is continuous in the weak$^*$ operator topology. Proof. Since $\pi$ is a representation, it follows that
$$
\begin{equation*}
\pi(\psi_1(h_1)\cdots\psi_n(h_n))= \pi(\psi_1(h_1))\cdots \pi(\psi_n(h_n))\quad\text{for each }\ h_i\in H_i.
\end{equation*}
\notag
$$
The mapping defined by
$$
\begin{equation*}
(h_1,\dots,h_n)\mapsto\pi(\psi_1(h_1))\cdots \pi(\psi_n(h_n)),
\end{equation*}
\notag
$$
where $h_i\in H_i$ for all $i=1,\dots,n$, is obviously separately continuous on the neighbourhood $V$ (mentioned in the assumption of the lemma) in the metrizable group $H=H_1\times \dots \times H_n$. We can assume that $V$ contains the closure of the product $W=\prod_{i=1}^n W_i$ of some open neighbourhoods $W_i\supset V_i$ of the identity element $e_i\in H_i$, $i=1, \dots,n$, with compact closures $\overline{W_i}$, $i=1, \dots,n$ (here and below, the bar means the closure with respect to the topology under consideration; in our case, this is the metrizable locally compact topology in $H_i$), on whose images the representation $\pi$ is continuous. Since every mapping $\pi\circ \psi_i$, $i=1, \dots,n$, is continuous on $W_i$, it follows that the image of the closure $\overline{W_i}$ of the neighbourhood $W_i$ under $\pi\circ\psi_i$ is compact in the space $\mathcal {L}(E)$ of continuous linear operators in the locally convex space $E$ of the representation $\pi$, where the space $\mathcal {L}(E)$ is equipped with the weak$^*$ operator topology. Since the multiplication of adjoint operators is separately continuous in the weak$^*$ operator topology on bounded sets, we see that the composed mapping
$$
\begin{equation*}
\pi\circ\psi\colon V\supset W_1\times \dots \times W_n\to G\to\mathcal{L}(E),
\end{equation*}
\notag
$$
defined by for is separately continuous in the weak$^*$ operator topology on the $\psi$-image of $W$, and the image of $W$ in $G$ is a neighbourhood of the identity element of $G$ due to the homeomorphism assumption.
By Namioka’s theorem [23], this mapping has a point of joint continuity in the image of $W$ in $G$. So, the representation $\pi$ of the group $G$ has points of weak$^*$ continuity. However, a representation which is weakly$^*$ continuous at one point is weakly$^*$ continuous on the whole group. Consider the case where $G$ is a connected Lie group. Let us choose a basis $e_1,\dots,e_n$ in the Lie algebra $\mathfrak g$ of the Lie group $G$ and consider the mapping of the finite-dimensional vector space ${\mathbb R}^n$ into $G$ defined by
$$
\begin{equation*}
\mathbb R^n\ni t=(t_1,\dots,t_n)\mapsto g(t)=\exp(t_1e_1)\cdots\exp(t_ne_n)\in G.
\end{equation*}
\notag
$$
This mapping is smooth and has the identity Jacobi matrix at zero; therefore, it defines a diffeomorphism of some open neighbourhood of the zero in $\mathbb{R}^n$ onto an open neighbourhood of the identity element in $G$. In particular, the set of one-parameter groups satisfies the conditions of the first two paragraphs of the lemma for $H=\mathbb R^n$.
This completes the proof of Lemma 2. The following result is immediate from the proof of Lemma 2. Corollary 1. Let $G$ be a Lie group and let $\pi$ be a (group-theoretic) representation of $G$ in a Fréchet space dual to a locally convex space. If the representation $\pi$ is continuous in the weak$^*$ operator topology on every one-parameter subgroup of the group $G$, then it is continuous in the weak$^*$ operator topology. The next auxiliary result is the most important technical tool in the proof of the main result. Lemma 3. Let $E$ be a Fréchet space dual to a locally convex space $E_*$, let $G$ be a topological group, and let $\pi$ be a locally uniformly bounded (group-theoretic) representation of the group $G$ on $E$ by adjoint continuous linear operators. If $\omega^*(\pi)<1$, then $\omega^*(\pi)=0$, and thus the representation $\pi$ is continuous in the weak$^*$ operator topology. Proof. Let $M$ be a closed bounded (and hence equicontinuous (see Theorem III.4.2 in [20]) subset of $\mathcal {L}(E)$ in the weak$^*$ operator topology. Assume that $M$ is convex and contains the zero element of $\mathcal {L}(E)$ (replacing $M$ by the bipolar of the set $M\cup\{0\}$ if necessary, where $M$ is closed and bounded, we see that this bipolar is closed and bounded (see Theorem 6.3.3 in [25] and the corollary in § III.3.4 of [20]) and convex as the intersection of convex sets).
The operator space $\mathcal {L}(E)$ admits a natural embedding $\mathcal {L}(E)\ni T\mapsto(T\xi)(\varphi)$, $\xi\in E$, $\varphi\in E_*$, in the product space $F^{E\times E_*}$, where $F$ is the ground field of $E$, and thus $\mathcal {L}(E)$ can be regarded as a subset of $F^{E\times E_*}$. By the very definition of the weak$^*$ operator topology on $\mathcal {L}(E)$, this topology coincides with the restriction to $\mathcal {L}(E)$ of the product topology on the product space $F^{E\times E_*}$. By Tikhonov’s Theorem, the compactness in $F^{E\times E_*}$, is equivalent to the conditions of being closed and pointwise bounded. Thus, to prove that a subset $M\subset\mathcal {L}(E)$ is compact, it suffices to show that $M$, which is equicontinuous, is pointwise bounded in $F^{E\times E_*}$ and also closed in this larger space. Since $M$ is equicontinuous, this implies immediately the desired pointwise boundedness. We claim that the closure $\overline{M}$ of $M$ in $F^{E\times E_*}$ is actually contained in $\mathcal {L}(E)$. Let $U\subset E$ be a $0$-neighbourhood equal to the polar of a subset $\mathring{U}\subset E_*$. Using the equicontinuity of $M$ again, we choose a $0$-neighbourhood $V\subset E$ such that $MV\subset U$. This means that, for $\xi\in V$, $\varphi\in \mathring{U}$, and $T\in M$, we have and this property remains valid for all elements of the closed set $\overline{M}$. We conclude that $TV\subseteq U$ holds for each $T\in \overline{M}$, and hence $\overline{M}\subseteq\mathcal{L}(E)$.This proves that the closed bounded (and hence equicontinuous) subsets $M\subset\mathcal{L}(E)$ are also closed in $F^{E\times E_*}$. The author is indebted to the referee for the above proof of this fact. Let now $M$ be the intersection of the closures $\overline{\pi(V)}$ of the images $\pi(V)$, where $V$ ranges over all neighbourhoods $V$ of the identity element in the group $G$; here, the bar means the closure of a set in the topology of $F^{E\times E_*}$. Since the representation $\pi$ is bounded on some neighbourhood $V_0$ of the identity element, it follows that $M$ is an equicontinuous subset of $\mathcal{L}(E)$ and, for the neighbourhoods $V\subset V_0$, the sets $\overline{\pi(V)}$ (closed in the topology of $F^{E\times E_*}$) are compact in the topology of $F^{E\times E_*}$ as the closed subsets of the compact set $\overline{\pi(V_0)}$ in the topology of $F^{E\times E_*}$. Thus, $M$ is a compact set in the topology of $F^{E\times E_*}$. Let $T$ be some element of $M$; we claim that $T=1_E$. Since for every neighbourhood $V\subset V_0$ of the identity element in $G$, for every neighbourhood $U$ of the zero element in $E$, for its polar $\mathring{U}$ in $E_*$, and for any vector $\xi$ in $U$ and any element $\varphi\in\mathring{U}$, we have for each $S\in \overline{\pi(V)}$, each $\xi\in U$, and each (real) functional $\varphi\in\mathring{U}$. By the definition of the operator $T$ and the set $M$, this yields the inequality $|(T\xi-\xi)(\varphi)|\leqslant q$ for each $\xi\in U$ and each functional $\varphi\in\mathring{U}$.For every neighbourhood $V$ of the identity element in $G$ and each positive integer $n$, there is a neighbourhood $W$ of the identity element in $G$ satisfying $W^{2^n}\subset V$. It follows that $\pi(W)^{2^n} \subset \pi(V)$, and, obviously $(\pi(W))^{2^n}\subset \pi(V)\subset{\overline {\pi(V)}}$. Let $n=1$. Then $\pi(W)\pi(W)\subset \pi(V)$ and, since the multiplication is separately continuous in the weak$^*$ operator topology on bounded sets, the obvious relation $B\pi(W)\subset \pi(V)$ implies that $B\overline{\pi(W)}\subset \overline{\pi(V)}$ for each $B\in \pi(W)$, which shows that $\pi(W)\overline{\pi(W)}\subset \overline{\pi(V)}$. Hence $\overline{\pi(W)} \overline{\pi(W)}\subset \overline{\pi(V)}$. By induction, this implies that, for any neighbourhood $V$ of the identity element in $G$, for any positive integer $n$, and for every neighbourhood $W$ of the identity element in $G$ satisfying $W^{2^n}\subset V$, we have $\{\overline{\pi(W)}\}^{2^n}\subset \overline{\pi(V)}$. In particular, since $T\in\overline{\pi(V)}$ for any neighbourhood $V$ of the identity element, it follows that $T\in\overline{\pi(W)}$, and hence $T^n\in \overline{\pi(V)}$ for each $n\in \mathbb N$ and for each neighbourhood $V$ of the identity element in $G$. Consequently, $T^n\in M$. As we saw above, this gives $|(T^n\xi-\xi)(\varphi)|\leqslant q$ for each $\xi\in U$, each functional $\varphi\in\mathring{U}$, and each positive integer $n$. Let $J$ be some invariant mean on $\mathbb N$. For any $\varphi\in E_*$ and any $\xi\in E$, we set $(B\xi)(\varphi)=J((T^n\xi)(\varphi))$ (a similar technique was used in a significantly more general situation in [25]). We thus obtain a continuous linear functional on $E_*$ (an element of $E$ absorbed by the bipolar of the neighbourhood $U$), that is, a vector $B\xi\in E$ linearly depending on $\xi\in E$ and such that
$$
\begin{equation*}
|(B\xi-\xi)(\varphi)|\leqslant q,\qquad \xi\in U,\quad \varphi\in\mathring{U}.
\end{equation*}
\notag
$$
In particular, $B\xi\neq0\in E$ for $0\neq\xi\in U$, since $|\xi(\varphi)|$ is arbitrarily close to 1 for $\xi\in U$ and $\varphi\in\mathring{U}$. In addition, from the invariance of the mean $J$ we have, for each $\varphi \in E_*$ and each $\xi\in E$,
$$
\begin{equation*}
(BT\xi)(\varphi)=J((T^{n}(T\xi))(\varphi)) =J((T^{n+1}\xi)(\varphi))=J((T^n\xi)(\varphi))=(B\xi)(\varphi),
\end{equation*}
\notag
$$
and so $B(T\xi)-B\xi=0$ in $E$.
Consequently, $B(T\xi-\xi)=0$ in $E$ if $\xi\neq0$ for $\xi\in U\subset E$, and $T\xi-\xi=0$ for each $\xi\in U$, since $B$ is injective, whence $T$ is the identity operator on $E$. Since the intersection of the closures $\overline{\pi(V)}$ in the topology of $F^{E\times E_*}$ of the sets $\pi(V)$ for all neighbourhoods $V$ of $e$ in $G$ is the singleton formed by the identity operator, it follows that the same holds for the closures of the same sets in the weak$^*$ operator topology. However, by definition of the weak$^*$ variation of the representation $\pi$, the equality $M=\{1_E\}$ in the weak$^*$ operator topology implies $\omega^*(\pi)=0$, and this also implies the continuity of the representation $\pi$ at $e$ in the weak$^*$ operator topology by assertion 2 of Lemma 1. This completes the proof of Lemma 3. The following result, which is a direct consequence of Lemma 3, is equivalent to Theorem 1. Theorem 2. Let $E$ be a Fréchet space dual to some locally convex space, $G$ be a topological group, and $\pi$ be a locally uniformly bounded (group-theoretic) representation of the group $G$ by adjoint operators in the space $E$. Assume that $\omega^*(\pi)<1$. Then $\omega^*(\pi)=0$, that is, the representation $\pi$ is continuous in the weak$^*$ operator topology. Corollary 2. Let $E$ be a Fréchet space dual to a locally convex space, let $\pi$ be a locally uniformly bounded (group-theoretic) representation of a topological group $G$ in $E$, and let there be a mapping $\rho$ of the group $G$ into $\mathcal {L}(E)$, continuous in the weak$^*$ operator topology, and such that the upper bound over all neighbourhoods $U$ of the zero element in $E$, all vectors $\xi\in U$, and all functionals $\varphi\in E_*$ in the polar $\mathring{U}$ of the neighbourhood $U$ of the lower bounds of the quantities $\sup_{g\in V}|(\pi(g)\xi- \rho(g)\xi)(\varphi)|$ over all neighbourhoods $V\subset G$ of the identity element $e\in G$ is less than $1/2$, Then the representation $\pi$ is continuous in the weak$^*$ operator topology. Proof. It follows immediately from the continuity of the mapping $\rho$ in the weak$^*$ operator topology, from the inequality in the condition of the corollary, and from Lemma 1 that $\omega(\pi)<1$. By Theorem 2, the representation $\pi$ is continuous in the weak$^*$ operator topology. This proves the corollary. AcknowledgementsThe author is greatly indebted to the anonymous referee, who indicated a lot of inaccuracies and misprints in the original version, and thus significantly helped to improve the text of the paper. 
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