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

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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.

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Acknowledgements

We would like to thank Sorin Popa for helpful comments.

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Authors and Affiliations

  1. Department of Mathematics, University of California San Diego, 9500 Gilman Drive, La Jolla, CA, 92093, USA

    Adrian Ioana & Matthew Wiersma

  2. Department of Mathematics, University of California Los Angeles, 520 Portola Plaza, Los Angeles, CA, 90095, USA

    Pieter Spaas

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A.I. was supported in part by NSF Career Grant DMS #1253402 and NSF FRG Grant #1854074.

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Ioana, A., Spaas, P. & Wiersma, M. Cohomological obstructions to lifting properties for full C\(^*\)-algebras of property (T) groups. Geom. Funct. Anal. 30, 1402–1438 (2020). https://doi.org/10.1007/s00039-020-00550-4

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