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 completely recognizable from their von Neumann algebras, L(G). Specifically, assume that G=A× W, where A is an infinite abelian group and W is an ICC wreath-like product group [CIOS22a; AMCOS23] with property (T) and trivial abelianization. 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 × H₀ where B is infinite abelian and H₀ is isomorphic to W. 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.