A note on some group $C^*$-algebras which are quasi-directly finite

TL;DR

This paper investigates the quasi-direct finiteness property in group $C^*$-algebras, extending known results from von Neumann algebras of discrete groups to more general locally compact groups.

Contribution

It generalizes the concept of quasi-direct finiteness to a broader class of group $C^*$-algebras, building on classical results and previous work on residual spectrum phenomena.

Findings

Group $C^*$-algebras of certain locally compact groups are quasi-directly finite.

Extension of Kaplansky's observation to a wider class of groups.

Connections to residual spectrum in convolution operators.

Abstract

An algebra is said to be quasi-directly finite when any left-invertible element in its unitization is automatically right-invertible. It is an old observation of Kaplansky that the von Neumann algebra of a discrete group has this property; in this note, we collate some analogous results for the group $C^*$-algebras of more general locally compact groups. Partial motivation comes from earlier work of the author on the phenomenon of empty residual spectrum for convolution operators.