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

TL;DR

This paper proves that reduced free group C*-algebras of different free group ranks are not isomorphic, using a novel approach involving embeddings into II$_1$ factors with freely independent Haar unitaries, providing a new proof of a classical result.

Contribution

It introduces a new method involving embedding spaces in II$_1$ factors to distinguish reduced free group C*-algebras of different ranks, offering a shorter proof of a known theorem.

Findings

$C^ ext{*}_r( ext{F}_n) cong C^ ext{*}_r( ext{F}_m)$ for $n eq m$

New embedding approach in II$_1$ factors used to differentiate C*-algebras

Recovers classical result with a simplified proof

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.