A ring $R$ is called Dedekind finite (or directly finite) if it contains no subsemigroup isomorphic to the bicyclic monoid. The ring $R$ is stably finite if all matrix algebras over $R$ are Dedekind finite. A famous theorem of Kaplansky from the late 1960s says that if $G$ is a group, then $KG$ is stably finite for any field $K$ of characteristic 0. Surprisingly his proof used $C^*$-algebras. Later proofs using more elementary analysis, were given but to the best of my knowledge all known proofs are analytic. Kaplansky asked whether group algebras are stably finite in positive characteristic and this question remains open. The best known result to date (due to Elek and Szabo) is that if $G$ is a sofic group (this includes all amenable and all residually finite groups - and possibly all groups), then $KG$ is stably finite for any field $K$ (and even any commutative ring $K$).

A natural question is: when is the semigroup algebra $KS$ of a semigroup $S$ stably finite? An obvious necessary condition is that $S$ contain no bicyclic subsemigroup. Is it sufficient? How about for inverse semigroups or cancellative semigroups? Munn proved in the nineties that if $S$ is a stable semigroup and each $J$-class of $S$ contains either finitely many $L$-classes or finitely many $R$-classes, then $KS$ is stably finite for any field $K$ of characteristic 0 using Kaplansky's result. This applies, for example to regular semigroups in which each $J$-class contains either finitely many $L$-classes or finitely many $R$-classes (e.g., to an inverse semigroup in which each $D$-class has finitely many idempotents). He also showed under his hypotheses that if every maximal subgroup of $S$ is abelian, then $KS$ is stably finite for any field $K$. He used here $PI$-properties matrices over commutative rings like the group ring of an abelian group.

In this talk, I show that under Munn's hypotheses, $KS$ is stably finite for a commutative ring $K$ iff $KG$ is stably finite for each maximal subgroup $G$. In particular, if every maximal subgroup of $S$ is stably finite, then $KS$ is stably finite for any commutative ring. If time permits, I'll talk about extensions to etale groupoid algebras.