Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-20T17:37:45.207Z Has data issue: false hasContentIssue false
  • English
  • Français

TRACES ON SEMIGROUP RINGS AND LEAVITT PATH ALGEBRAS

Part of: Rings and algebras arising under various constructions Rings and algebras with additional structure Semigroups

Published online by Cambridge University Press:  21 July 2015

ZACHARY MESYAN  and
LIA VAŠ
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]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

MSC classification

Primary: 16S36: Ordinary and skew polynomial rings and semigroup rings
Secondary: 16W10: Rings with involution; Lie, Jordan and other nonassociative structures 16S50: Endomorphism rings; matrix rings 20M25: Semigroup rings, multiplicative semigroups of rings
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 58 , Issue 1 , January 2016 , pp. 97 - 118
Copyright
Copyright © Glasgow Mathematical Journal Trust 2015 

References

REFERENCES

1.Abrams, G., Ara, P. and Siles Molina, M., Leavitt path algebras: a primer and handbook, (Springer, to appear).CrossRefGoogle Scholar
2.Abrams, G. and Aranda Pino, G., The Leavitt path algebra of a graph, J. Algebra 293 (2005), 319–334.CrossRefGoogle Scholar
3.Abrams, G., Aranda Pino, G., Perera, F. and Siles Molina, M., Chain conditions for Leavitt path algebras, Forum Math. 22 (2010), 95–114.CrossRefGoogle Scholar
4.Abrams, G. and Mesyan, Z., Simple lie algebras arising from Leavitt path algebras, J. Pure Appl. Algebra 216 (2012), 2302–2313.CrossRefGoogle Scholar
5.Ara, P., Moreno, M. A. and Pardo, E., Nonstable K-theory for graph algebras, Algebr. Represent. Theory 10 (2007), 157–178.CrossRefGoogle Scholar
6.Aranda Pino, G., Rangaswamy, K. L. and Vaš, L., *-regular Leavitt path algebra of arbitrary graphs, Acta Math. Sci. Ser. B Engl. Ed. 28 (2012), 957–968.CrossRefGoogle Scholar
7.Ash, C. J. and Hall, T. E., Inverse semigroups on graphs, Semigroup Forum 11 (1975), 140–145.CrossRefGoogle Scholar
8.Berberian, S. K., Baer *-rings, Die Grundlehren der mathematischen Wissenschaften, vol. 195 (Springer-Verlag, Berlin-Heidelberg-New York, 1972).CrossRefGoogle Scholar
9.Jones, D. G. and Lawson, M. V., Graph inverse semigroups: Their characterization and completion, J. Algebra 409 (2014), 444–473.CrossRefGoogle Scholar
10.Krieger, W., On subshifts and semigroups, Bull. London Math. Soc. 38 (2006), 617–624.CrossRefGoogle Scholar
11.Mesyan, Z., Commutator rings, Bull. Aust. Math. Soc. 74 (2006), 279–288.CrossRefGoogle Scholar
12.Mesyan, Z., Commutator Leavitt path algebras, Algebr. Represent. Theory 16 (2013), 1207–1232.CrossRefGoogle Scholar
13.Pask, D. and Rennie, A., The noncommutative geometry of graph C*-algebras I: The index theorem, J. Funct. Anal. 233 (2006), 92–134.CrossRefGoogle Scholar
14.Pask, D., Rennie, A. and Sims, A., Noncommutative manifolds from graph and k-graph C*-algebras, Commun. Math. Phys. 292 (2009), 607–636.CrossRefGoogle Scholar
15.Paterson, A. L. T., Graph inverse semigroups, groupoids and their C*-algebras, J. Operator Theory 48 (2002), 645–662.Google Scholar
16.Vaš, L., Canonical traces and directly finite Leavitt path algebras, Algebr. Represent. Theory (2015), doi: 10.1007/s10468-014-9513-8.CrossRefGoogle Scholar