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

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