Property (T) for Banach algebras

Abstract

We define and study the notion of property $(\rm T)$ for Banach algebras, generalizing the one from $C^*$-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 $(\rm{T}_{\mathcal E})$ of the symmetrized pseudofunction algebras $F^*_{\mathcal E}(G)$ characterizes the Banach property $(\rm{T}_{\mathcal E})$ 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 $1\leq p < \infty$, we also achieve the analogue characterization using the symmetrized $p$-pseudofunction algebras $F^*_{\lambda_ p}(G)$.