Property (T) for uniformly bounded representations and weak*-continuity of invariant means

TL;DR

This paper introduces a strengthened version of Kazhdan's Property (T) for uniformly bounded representations, characterizes it via invariant means, and explores its implications for Lie groups and affine actions on Hilbert spaces.

Contribution

It defines and systematically studies a new property strengthening Property (T), linking it to weak*-continuity of invariant means and analyzing its consequences for specific Lie groups.

Findings

Characterization of the strengthened Property (T) via weak*-continuity.

Invariant properties at the von Neumann algebra level.

Existence of proper affine actions for certain Lie groups.

Abstract

For every $c\geq 1$, we define a strengthening of Kazhdan's Property (T) by considering uniformly bounded representations $\pi$ with fixed bound $|\pi|\leq c$. We carry out a systematic study of this property and show that it can be characterised by the weak*-continuity of the unique invariant mean on a suitable space of coefficients. For countable groups, we prove that the family of properties thus obtained yield an invariant at the von Neumann algebra level. Moreover, by focusing on certain representations of rank 1 Lie groups, we show that $\operatorname{Sp}(n,1)$ and $F_{4,-20}$ admit proper uniformly Lipschitz affine actions on Hilbert spaces.