Topological centers of module actions induced by unitary representations
Let \(G\) be a locally compact group. Let \(LUC(G)\) denote the Banach space of all left uniformly continuous functions on \(G\). Then the dual space \(LUC(G)^*\) becomes a Banach algebra. Let \(\pi\) be a unitary representation of \(G\) on a complex Hilbert space \(H\). Then \(UCB(\pi)\) denotes the unital \(C^*\)-algebra of \(B(H)\) consisting of the operators \(T\in B(H)\) such that the map \(t\mapsto\pi(t^{-1})T\pi(t)\) is continuous. The Banach space \(UCB(\pi)^*\) becomes a Banach left \(LUC(G)^*\)-module in a natural way. The author studies the topological center of this module. A characterisation of when the center is minimal and also conditions for the center to be maximal are given. Various examples whose topologial centers are maximal, minimal or neither are also given.
scientific article; zbMATH DE number 5801188
scientific article