Cohomological obstructions to lifting properties for full C$$^*$$-algebras of property (T) groups

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 $$\mathbb {Z}^2\times \text {SL}_2({\mathbb {Z}})$$ 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 $$^*$$ -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.