Directly finite algebras of pseudofunctions on locally compact groups

TL;DR

This paper investigates conditions under which algebras of pseudofunctions on locally compact groups are directly finite, extending known results for group C*-algebras and exploring implications for specific groups.

Contribution

It establishes that reduced group C*-algebras of unimodular groups are directly finite and characterizes when algebras of p-pseudofunctions are directly finite based on group properties.

Findings

Reduced group C*-algebras of unimodular groups are directly finite.

Algebras of p-pseudofunctions are directly finite if G is amenable and unimodular.

L^1(G) is not directly finite for affine groups of real or complex lines.

Abstract

An algebra $A$ is said to be directly finite if each left invertible element in the (conditional) unitization of $A$ is right invertible. We show that the reduced group ${\rm C}^\ast$-algebra of a unimodular group is directly finite, extending known results for the discrete case. We also investigate the corresponding problem for algebras of $p$-pseudofunctions, showing that these algebras are directly finite if $G$ is amenable and unimodular, or unimodular with the Kunze--Stein property. An exposition is also given of how existing results from the literature imply that $L^1(G)$ is not directly finite when $G$ is the affine group of either the real or complex line.