Journal of Operator Theory

Volume 94, Issue 2, Autumn 2025 pp. 403-426.

Property $(\mathrm{T})$ for Banach algebras

Authors: Emilie Mai Elkiaer (1), Sanaz Pooya (2)
Author institution: (1) Department of Mathematics, University of Oslo, Oslo, 0851, Norway
(2) Institute of Mathematics, University of Potsdam, Potsdam, 14476, Germany

Summary: 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 finite $1\leqslant p$, we also achieve the analogue characterization using the symmetrized $p$-pseudofunction algebras $F^*_{\lambda_ p}(G)$.

DOI: http://dx.doi.org/10.7900/jot.2023dec11.2463
Keywords: Property $(\rm T)$, property $(\mathrm{T}_{L^p})$, Banach algebras, pseudofunction algebras, locally compact groups

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