Abstract:
A $C^*$-algebra has the lifting property (LP) if any unital completely positive map into a quotient $C^*$-algebra admits completely positive lift. The local lifting property (LLP), introduced by Kirchberg in the early 1990s, is a weaker, local version of the LP.
      In this talk, I will present 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). This allows us to derive that the full $C^*$-algebras of the groups $Z^{2}\rtimes SL_2(Z)$ and $SL_n(Z)$, for $n>2$, do not have the LLP. We also prove that the full $C^*$-algebras of a large class of groups with property (T), including those admitting a probability measure preserving action with non-vanishing second real-valued cohomology, do not have the LP. Time permitting, I will also discuss a connection with the notion of Hilbert-Schmidt stability for countable groups. This is based on joint work with Pieter Spaas and Matthew Wiersma.