Rigidity results for group von Neumann algebras with diffuse center

We introduce the first examples of groups G with infinite center which in a natural sense are recognizable from their von Neumann algebras, L(G). Specifically, assume that G = A x G_0, where A is an infinite abelian group and G_0 is an ICC wreath-like product group with property (T), trivial abelianization and torsion free outer automorphism group. Then whenever H is an arbitrary group such that L(G) is *-isomorphic to L(H), via an arbitrary *-isomorphism preserving the canonical traces, it must be the case that H = B x H_0 where B is infinite abelian and H_0 is isomorphic to G_0. Moreover, we completely describe the *-isomorphism between L(G) and L(H). This yields new applications to the classification of group C*-algebras, including examples of non-amenable groups which are recoverable from their reduced C*-algebras but not from their von Neumann algebras. This is joint work with Ionut Chifan and Hui Tan.