Cohomological obstructions to lifting properties for full group C$^*$-algebras

TL;DR

This paper introduces a cohomological method to demonstrate the failure of lifting properties in full group C*-algebras, especially for groups with property (T), expanding understanding of their algebraic obstructions.

Contribution

It develops a new cohomology-based approach to identify when full group C*-algebras lack lifting properties, including for groups with property (T) and non-vanishing second cohomology.

Findings

Full C*-algebras of certain groups lack the local lifting property (LLP).

Full C*-algebras of groups with property (T) and non-zero second cohomology lack the lifting property (LP).

Non-finitely presented property (T) groups' full C*-algebras fail the LP.

Abstract

We develop a new method, based on non-vanishing of second cohomology groups, for proving the failure of lifting properties for full C$^*$-algebras of countable groups with (relative) property (T). We derive that the full C$^*$-algebras of the groups $\mathbb Z^2\rtimes\text{SL}_2(\mathbb Z)$ and $\text{SL}_n(\mathbb Z)$, for $n\geq 3$, do not have the local lifting property (LLP). We also prove that the full C$^*$-algebras of a large class of groups $\Gamma$ with property (T), including those such that $\text{H}^2(\Gamma,\mathbb R)\not=0$ or $\text{H}^2(\Gamma,\mathbb Z\Gamma)\not=0$, 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$^*$-algebra of any non-finitely presented property (T) group fails the LP.