Directly finite algebras of pseudofunctions on locally compact groups

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 $C^*$-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