$B^p_r(F_n)$ has no nontrivial idempotents

TL;DR

This paper proves that the reduced group algebra of a free group has no nontrivial idempotents, revealing a fundamental algebraic property of these operator algebras.

Contribution

It establishes the absence of nontrivial idempotents in the algebra of free groups for all positive integers n, a new result in operator algebra theory.

Findings

No nontrivial idempotents in algebra of free groups

Results hold for all positive integers n

Advances understanding of algebraic structure of free group algebras

Abstract

We show that there is no nontrivial idempotent in the reduced group $\ell^p$-operator algebra $B^p_r(F_n)$ of the free group $F_n$ on $n$ generators for each positive integer $n$.