  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.
