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TRACES ON SEMIGROUP RINGS AND LEAVITT PATH ALGEBRAS

Published online by Cambridge University Press:  21 July 2015

ZACHARY MESYAN
Affiliation:
Department of Mathematics, University of Colorado, Colorado Springs, CO 80918, USA e-mail: [email protected]
LIA VAŠ
Affiliation:
Department of Mathematics, Physics and Statistics, University of the Sciences, Philadelphia, PA 19104, USA e-mail: [email protected]
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Abstract

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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.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 2015 

References

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