Weak⁎-continuity of invariant means on spaces of matrix coefficients

Abstract

With every locally compact group G, one can associate several interesting bi-invariant subspaces X(G) of the weakly almost periodic functions WAP(G) on G, each of which captures parts of the representation theory of G. Under certain natural assumptions, such a space X(G) carries a unique invariant mean and has a natural predual, and we view the weak⁎-continuity of this mean as a rigidity property of G. Important examples of such spaces X(G), which we study explicitly, are the algebra McbAp(G) of p-completely bounded multipliers of the Figà-Talamanca–Herz algebra Ap(G) and the p-Fourier–Stieltjes algebra Bp(G). In the setting of connected Lie groups G, we relate the weak⁎-continuity of the mean on these spaces to structural properties of G. Our results generalise results of Bekka, Kaniuth, Lau and Schlichting.