Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Juan Felipe Ariza Mejía, Ionuţ Chifan, Adriana Fernández Quero

TL;DR

The paper addresses the problem of recovering a countable group from operator-algebraic invariants, contrasting the information retained by reduced $C^*$-algebras $C_r^*(G)$ versus group von Neumann algebras $L(G)$. It develops a framework around wreath-like and amalgamated free product groups forming the class $ ext{AT}$, establishing strong $W^*$-rigidity for these groups and product-structure rigidity for their direct products, while showing that infinite direct sums can fail $W^*$-rigidity due to McDuff phenomena. It further proves $C^*$-rigidity results for infinite direct sums of torsion-free $ ext{AT}$ groups, and shows that certain $ ext{AT}_0$ groups have endomorphisms of $C_r^*(G)$ that are all weakly inner; it also provides a continuum of examples with nonembedding properties in the $ ext{AT}_1$ subclass. The results reveal a sharp dichotomy: nonamenable groups can be $C_r^*(G)$-superrigid without being $L(G)$-superrigid, and place $C_r^*(G)$-based symmetries on a finer footing than von Neumann-algebraic symmetries, with broad implications for rigidity in group theory and operator algebras.

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.

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

TL;DR

The paper addresses the problem of recovering a countable group from operator-algebraic invariants, contrasting the information retained by reduced -algebras versus group von Neumann algebras . It develops a framework around wreath-like and amalgamated free product groups forming the class , establishing strong -rigidity for these groups and product-structure rigidity for their direct products, while showing that infinite direct sums can fail -rigidity due to McDuff phenomena. It further proves -rigidity results for infinite direct sums of torsion-free groups, and shows that certain groups have endomorphisms of that are all weakly inner; it also provides a continuum of examples with nonembedding properties in the subclass. The results reveal a sharp dichotomy: nonamenable groups can be -superrigid without being -superrigid, and place -based symmetries on a finer footing than von Neumann-algebraic symmetries, with broad implications for rigidity in group theory and operator algebras.

Abstract

Using techniques at the intersection of deformation/rigidity theory, geometric group theory, and the theory of -algebras, we construct a continuum of nonamenable groups that can be completely reconstructed from their reduced -algebras , but not from their group von Neumann algebras . 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 -superrigid and -superrigid. Finally, for a fairly large subclass of these amalgamated free product groups , we show that all -endomorphisms of are weakly inner.
Paper Structure (15 sections, 34 theorems, 54 equations)

This paper contains 15 sections, 34 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Outline of the paper.
  4. Acknowledgments.
  5. Preliminaries on wreath-like product groups
  6. Preliminaries on von Neumann algebras
  7. Intertwining-by-bimodules
  8. Amalgamated free product of von Neumann algebras
  9. Cartan subalgebras and equivalence relations
  10. Strong rigidity for orbit equivalence embeddings
  11. W*-superrigidity results for groups in class AT
  12. Product and embedding rigidity
  13. Reconstruction of group direct product splitting under W* equivalence
  14. W* and C* superrigidity for product groups
  15. Automorphisms and embeddings

Key Result

Theorem 1.1

There exists a family of countable groups $\{G_i\}_{i \in I}$ with $|I| = 2^{\aleph_0}$ such that $G_i \not\cong G_j$ whenever $i \neq j$, and each group $G_i$ is $C^*$-superrigid but not $W^*$-superrigid.

Theorems & Definitions (74)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Definition 2.1
  • Theorem 2.2: CIOS5
  • ...and 64 more