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Non-isomorphism of reduced free group $C^\ast$-algebras

David Gao, Srivatsav Kunnawalkam Elayavalli

TL;DR

This work addresses the nonisomorphism problem for reduced free group C*-algebras $C_r^*(\mathbb{F}_n)$ across different $n$. It develops a novel, K-theory–free method by embedding these algebras into a II$_1$ factor $M$ with freely independent Haar unitaries and analyzing the embedding space $\mathrm{Emb}(A,M)$ via topological and homotopy techniques. Central to the approach is showing $\mathrm{Emb}(A,M)$ is a weak homotopy equivalent to $U(M)^n$ and using the fundamental group of the unitary group, $\pi_1(U(M))\cong \mathbb{R}$, to deduce that $n$ must equal $m$ if $C_r^*(\mathbb{F}_n)\cong C_r^*(\mathbb{F}_m)$. The results provide a conceptually different proof from the classical Pimsner–Voiculescu argument and highlight the role of free independence in II$_1$ factors as a tool for distinguishing C*-algebraic invariants. The method aligns with broader trends linking free probability, ultrapower techniques, and operator-algebraic topology.

Abstract

Using a new approach involving embedding spaces in II$_1$ factors with plenty of freely independent Haar unitaries, we prove that $C^\ast_r(\mathbb{F}_n)\ncong C^\ast_r(\mathbb{F}_m)$ for $n \neq m$. This recovers the seminal result of Pimsner and Voiculescu with a short new proof.

Non-isomorphism of reduced free group $C^\ast$-algebras

TL;DR

This work addresses the nonisomorphism problem for reduced free group C*-algebras across different . It develops a novel, K-theory–free method by embedding these algebras into a II factor with freely independent Haar unitaries and analyzing the embedding space via topological and homotopy techniques. Central to the approach is showing is a weak homotopy equivalent to and using the fundamental group of the unitary group, , to deduce that must equal if . The results provide a conceptually different proof from the classical Pimsner–Voiculescu argument and highlight the role of free independence in II factors as a tool for distinguishing C*-algebraic invariants. The method aligns with broader trends linking free probability, ultrapower techniques, and operator-algebraic topology.

Abstract

Using a new approach involving embedding spaces in II factors with plenty of freely independent Haar unitaries, we prove that for . This recovers the seminal result of Pimsner and Voiculescu with a short new proof.
Paper Structure (2 sections, 7 theorems, 7 equations)

This paper contains 2 sections, 7 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Proof of main result

Key Result

Lemma 2.2

Let $\{a_i\}_{i \in I}$ be a set of generators of $A$. Then the map $\iota: \text{Emb}(A, M) \ni \pi \mapsto (\pi(a_i))_{i \in I} \in M^I$ is a topological embedding.

Theorems & Definitions (13)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • Theorem 2.7
  • ...and 3 more