Vector-valued invariant means on spaces of bounded operators associated to a locally compact group

Abstract

The purpose of this paper is to introduce and study the notion of a vector-valued $\pi\text{-invariant}$ mean associated to a unitary representation $\pi$ of a locally compact group $G$ on $\mathcal{S},$ a self-adjoint linear subspace containing $I$ of $\mathcal{B}(H_\pi).$ We obtain, among other results, an extension theorem for $\pi\text{-invariant}$ completely positive maps and $\pi\text{-invariant}$ means which characterizes amenability of $G.$ We also study vector-valued means on $\mathcal{S}$ of $\pi\text{-(weakly)}$ almost periodic operators on $H_\pi.$