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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.