2020/06/02 by Ioana, Adrian, Spaas, Pieter, Wiersma, Matthew
#FOS: Mathematics #Functional Analysis (math.FA) #Group Theory (math.GR) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2006.01874
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 Z2\rtimesSL2(\mathbb Z) and SLn(\mathbb Z), for n≥ 3, do not have the local lifting property (LLP). We also prove that the full C^*-algebras of a large class of groups Γ with property (T), including those such that H2(Γ,\mathbb R)\not=0 or H2(Γ,\mathbb ZΓ)\not=0, do not have the lifting property (LP). More generally, we show that the same holds if Γ 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.