On the set of topologically invariant means on an algebra of convolution operators on $L^p(G)$

Abstract:

Let $G$ be a locally compact group, $A_{p}=A_{p}(G)$ the Banach algebra defined by Herz; thus $A_{2}(G)=A(G)$ is the Fourier algebra of $G$. Let $PM_{p}=A^{*}_{p}$ the dual, $J \subset A_{p}$ a closed ideal, with zero set $F=Z(J)$, and $\mathbb {P} = (A_{p}/J)^{*}$. We consider the set $TIM_{\mathbb {P}}(x) \subset {\mathbb {P}}^{*}$ of topologically invariant means on $\mathbb {P}$ at $x\in F$, where $F$ is “thin.” We show that in certain cases card $TIM_{\mathbb {P}}(x) \geq 2^{c}$ and $TIM_{\mathbb {P}}(x)$ does not have the WRNP, i.e. is far from being weakly compact in $\mathbb {P}^{*}$. This implies the non-Arens regularity of the algebra $A_{p}/J$.