In the 30's F. Murray and J. von Neumann found a natural way to associate a von Neumann algebra $L(G)$ to every countable group $G$. The problem of classifying $L(G)$ in terms of $G$ emerged as a natural but highly challenging endeavor. Von Neumann algebras tend to have very limited recollection of the underlying group---perhaps best illustrated by A. Connes' celebrated result ('76) asserting that all amenable groups with infinite nontrivial conjugacy classes (ICC) yield isomorphic von Neumann algebras. Consequently, in such cases, $L(G)$ does not retain any additional information on $G$ besides amenability. In the non-amenable case, the classification problem is wide-open and far more intricate. Over the past 10-15 years, Popa deformation/rigidity theory has unveiled instances when $L(G)$ completely retains certain algebraic group properties of $G$. However, these results have predominantly focused on the case when the underlying groups $G$ are ICC. In this talk we expand this line of inquiry, by introducing the first examples of non-amenable groups $G$ with infinite center which, in a natural sense, are recognizable from their von Neumann algebras, $L(G)$. Specifically, assume that $G = A \times W$, where $A$ is an infinite abelian group and $W$ 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 $\ast$-isomorphic to $L(H)$, via an arbitrary $\ast$-isomorphism preserving the canonical traces, it must be the case that $H = B\times V$ where $B$ is an infinite abelian group and $V$ is a group isomorphic to $W$. Furthermore, we provide a complete description of the $\ast$-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 based on a recent joint work Adriana Fernandez I Quero and Hui Tan.