Cohomological obstructions to lifting properties for full group C${}^{\ast }$-algebras

Abstract

We develop a new method, based on non-vanishing of second cohomology groups, for proving the failure of lifting properties for full C${}^{\ast }$-algebras of countable groups with (relative) property (T). We derive that the full C${}^{\ast }$-algebras of the groups ${\mathbb{Z}}^2⋊{\text{SL}}_2(\mathbb{Z})$ and ${\text{SL}}_n(\mathbb{Z})$, for $n\ge 3$, do not have the local lifting property (LLP). We also prove that the full C${}^{\ast }$-algebras of a large class of groups $\Gamma$ with property (T), including those such that ${\text{H}}^2(\Gamma ,\mathbb{R})\ne 0$ or ${\text{H}}^2(\Gamma ,\mathbb{Z}\Gamma )\ne 0$, do not have the lifting property (LP). More generally, we show that the same holds if $\Gamma$ admits a probability measure preserving action with non-vanishing second $\mathbb{R}$-valued cohomology. Finally, we prove that the full C${}^{\ast }$-algebra of any non-finitely presented property (T) group fails the LP.

References in corpus (1)

• Strong solidity of group factors from lattices in SO(n,1) and SU(n,1)