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Cohomological obstructions to lifting properties for full C $$^*$$ ∗ -algebras of property (T) groups
Geometric and Functional Analysis  (IF2.148),  Pub Date : 2020-10-26, DOI: 10.1007/s00039-020-00550-4
Adrian Ioana, Pieter Spaas, Matthew Wiersma

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.