Non-amenable C$^*$-superrigid groups that are not W$^*$-superrigid

TL;DR

This paper constructs non-amenable groups that are uniquely determined by their reduced $C^*$-algebras but not by their von Neumann algebras, revealing new rigidity phenomena in operator algebras.

Contribution

It introduces the first examples of non-amenable groups with $C^*$-superrigidity without $W^*$-superrigidity, using advanced deformation and geometric techniques.

Findings

Existence of non-amenable groups reconstructed from $C^*$-algebras but not von Neumann algebras

Examples of groups that are both $C^*$- and $W^*$-superrigid

All $ ext{*$-}$-endomorphisms of certain $C^*$-algebras are weakly inner

Abstract

Using techniques at the intersection of deformation/rigidity theory, geometric group theory, and the theory of $C^*$-algebras, we construct a continuum of nonamenable groups $G$ that can be completely reconstructed from their reduced $C^*$-algebras $C_r^*(G)$, but not from their group von Neumann algebras $\mathrm{L}(G)$. These groups arise as infinite direct sums of amalgamated free product groups and constitute the first known examples of nonamenable groups exhibiting this phenomenon. In addition, we provide examples of finite direct products of amalgamated free product groups that are simultaneously $C^*$-superrigid and $W^*$-superrigid. Finally, for a fairly large subclass of these amalgamated free product groups $G$, we show that all $\ast$-endomorphisms of $C_r^*(G)$ are weakly inner.