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

TL;DR

This paper investigates the weak*-continuity of invariant means on certain subspaces of weakly almost periodic functions on groups, linking this property to the structural features of the groups, especially connected Lie groups.

Contribution

It establishes the weak*-continuity of invariant means on specific bi-invariant subspaces and relates this to the structural properties of connected Lie groups, generalizing previous results.

Findings

Weak*-continuity of invariant means on $X(G)$ spaces is established.

Connections between mean continuity and group structure are demonstrated.

Results extend known theorems to broader classes of groups.

Abstract

With every locally compact group $G$, one can associate several interesting bi-invariant subspaces $X(G)$ of the weakly almost periodic functions $\mathrm{WAP}(G)$ 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 $M_{\mathrm{cb}}A_p(G)$ of $p$-completely bounded multipliers of the Fig\`a-Talamanca-Herz algebra $A_p(G)$ and the $p$-Fourier-Stieltjes algebra $B_p(G)$. 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.