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On $ C^*$-algebras associated to right LCM semigroups


Authors: Nathan Brownlowe, Nadia S. Larsen and Nicolai Stammeier
Journal: Trans. Amer. Math. Soc. 369 (2017), 31-68
MSC (2010): Primary 46L05; Secondary 20M10, 20M30, 46L55
DOI: https://doi.org/10.1090/tran/6638
Published electronically: March 9, 2016
MathSciNet review: 3557767
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Abstract: We initiate the study of the internal structure of $ C^*$-algebras associated to a left cancellative semigroup in which any two principal right ideals are either disjoint or intersect in another principal right ideal; these are variously called right LCM semigroups or semigroups that satisfy Clifford's condition. Our main findings are results about uniqueness of the full semigroup $ C^*$-algebra. We build our analysis upon a rich interaction between the group of units of the semigroup and the family of constructible right ideals. As an application we identify algebraic conditions on $ S$ under which $ C^*(S)$ is purely infinite and simple.


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Additional Information

Nathan Brownlowe
Affiliation: School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, New South Wales 2522, Australia
Email: nathanb@uow.edu.au

Nadia S. Larsen
Affiliation: Department of Mathematics, University of Oslo, P. O. Box 1053, Blindern, 0316 Oslo, Norway
Email: nadiasl@math.uio.no

Nicolai Stammeier
Affiliation: Mathematisches Institut, Westfälischen Wilhelms-Universität Münster, 48149 Münster, Germany
Address at time of publication: Department of Mathematics, University of Oslo, P. O. Box 1053, Blindern, 0316 Oslo, Norway
Email: nicolsta@math.uio.no

DOI: https://doi.org/10.1090/tran/6638
Received by editor(s): June 29, 2014
Received by editor(s) in revised form: November 25, 2014
Published electronically: March 9, 2016
Additional Notes: Part of this research was carried out while all three authors participated in the workshop “Operator algebras and dynamical systems from number theory” in November 2013 at the Banff International Research Station, Canada. They thank BIRS for its hospitality and excellent working environment. The third author was supported by DFG through SFB $878$ and by ERC through AdG $267079$.
Article copyright: © Copyright 2016 American Mathematical Society