Traces on Semigroup Rings and Leavitt Path Algebras

TL;DR

This paper studies and classifies traces on semigroup rings and Leavitt path algebras, unifying various examples of linear functions that vanish on commutators and exploring their minimal and faithful traces.

Contribution

It generalizes the concept of traces to semigroup rings and Leavitt path algebras, providing classification and existence results for minimal and faithful traces.

Findings

Every semigroup ring admits a minimal trace.

Complete classification of minimal traces on these rings.

Identification of conditions for faithful traces on Leavitt path algebras.

Abstract

The trace on matrix rings, along with the augmentation map and Kaplansky trace on group rings, are some of the many examples of linear functions on algebras that vanish on all commutators. We generalize and unify these examples by studying traces on (contracted) semigroup rings over commutative rings. We show that every such ring admits a minimal trace (i.e., one that vanishes only on sums of commutators), classify all minimal traces on these rings, and give applications to various classes of semigroup rings and quotients thereof. We then study traces on Leavitt path algebras (which are quotients of contracted semigroup rings), where we describe all linear traces in terms of central maps on graph inverse semigroups and, under mild assumptions, those Leavitt path algebras that admit faithful traces.