A note on some group $C^{\ast }$-algebras which are quasi-directly finite

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 $C^{\ast }$-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.

v1: 10 pages, preliminary version. v2: 9 pages. New references and expanded discussion of the connected case, correcting an error in the previous version; other minor improvements throughout