Property (T) for Banach algebras

Abstract

We define and study the notion of property $(\mathrm{T})$ for Banach algebras, generalizing the one from $C^{\ast }$-algebras. For a second countable locally compact group $G$ and a given family of Banach spaces $\mathcal{E}$, we prove that our Banach algebraic property $({\mathrm{T}}_{\mathcal{E}})$ of the symmetrized pseudofunction algebras $F_{\mathcal{E}}^{\ast }(G)$ characterizes the Banach property $({\mathrm{T}}_{\mathcal{E}})$ of Bader, Furman, Gelander and Monod for groups. In case $G$ is a discrete group and $\mathcal{E}$ is the class of $L^p$-spaces for $1\le p<\infty$, we also achieve the analogue characterization using the symmetrized $p$-pseudofunction algebras $F_{\hat{I}{»}_p}^{\ast }(G)$.

20 pages; v2: accepted version to appear in Journal of Operator Theory