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

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.