On topological centers induced by unitary representations
Topological centers is a well-known notion and has been studied by many authors. In [\textit{P.-K. Chan}, J. Funct. Anal. 259, No. 9, 2193--2214 (2010; Zbl 1214.46030)], the notion of ``topological centers induced by unitary representations'' was introduced. The aim of this article is to study the case when the topological center induced by a unitary representation is the largest possible one. Let \(G\) be a locally compact group and \(\pi:G\to U(H)\) be a continuous unitary representation of \(G\). Denote by \(LUC(G)\) the space of all left uniformly continuous functions on \(G\). Consider \begin{itemize} \item \(UCB(\pi):=\{T\in \mathcal{B}(H): s\mapsto \pi_s^*T\pi_s \text{ is a norm continuous map from }G\text{ to }\mathcal{B}(H)\}\); \item \(WAP(\pi):= \{T\in \mathcal{B}(H): \{\pi_s^*T\pi_s:s\in G\} \text{ is relatively weakly compact}\}\). \end{itemize} Then \(WAP(\pi) \subseteq UCB(\pi)\). If \(\varphi\in UCB(\pi)^*\) and \(T\in UCB(\pi)\), it was shown in [\textit{P.-K. Chan}, J. Funct. Anal. 259, No. 9, 2193--2214 (2010; Zbl 1214.46030)] that the function \(\varphi T:G\to \mathbb{C}\) given by \[\varphi T(s):= \varphi(\pi_s^*T\pi_s)\qquad (s\in G)\] belongs to \(LUC(G)\). Thus, for \(m\in LUC(G)^*\) and \(\varphi\in UCB(\pi)^*\), one can define an element \(m\cdot \varphi\in UCB(\pi)^*\) by \((m\cdot \varphi)(T) = m(\varphi T)\), for \(T\in UCB(\pi)\). The \textit{topological center of the module action induced by \(\pi\)}, denoted by \(Z(\pi)\), is the collection of those \(m\in LUC(G)^*\) such that the map \(\varphi \mapsto m\cdot \varphi\) from \(UCB(\pi)^*\) to itself is weak\(^*\)-weak\(^*\)-continuous. In general, \(Z(\pi)\) is a Banach subalgebra of \(LUC(G)^*\). The main result of this article is the following. For a continuous unitary representation \((\pi,H)\) of \(G\), one has \(Z(\pi) = LUC(G)^*\) if and only if \(WAP(\pi) = UCB(\pi)\). In order to prove this result, the author gives a description of elements in \(WAP(\pi)\) amongst elements \(T\in UCB(\pi)\), via weak compactness of a canonical map \(\gamma_T:L^1(G)\to UCB(\pi)\) induced by \(T\). Moreover, as an application of the main result, the author also obtains the following: A locally compact group \(G\) is compact if and only if \(WAP(\lambda) = UCB(\lambda)\), where \(\lambda\) is the left regular representation of \(G\).
scientific article; zbMATH DE number 7378006
scientific article