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JOINS AND COVERS IN INVERSE SEMIGROUPS AND TIGHT ${C}^{\ast } $-ALGEBRAS

Part of: Selfadjoint operator algebras Groups and semigroups of linear operators, their generalizations and applications Semigroups

Published online by Cambridge University Press:  10 January 2014

ALLAN P. DONSIG  and
DAVID MILAN
ALLAN P. DONSIG*
Affiliation:
Mathematics Department, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, USA
DAVID MILAN
Affiliation:
Department of Mathematics, University of Texas at Tyler, Tyler, TX 75799, USA email [email protected]
*
✉[email protected]
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Abstract

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We show Exel’s tight representation of an inverse semigroup can be described in terms of joins and covers in the natural partial order. Using this, we show that the ${C}^{\ast } $-algebra of a finitely aligned category of paths, developed by Spielberg, is the tight ${C}^{\ast } $-algebra of a natural inverse semigroup. This includes as a special case finitely aligned higher-rank graphs: that is, for such a higher-rank graph $\Lambda $, the tight ${C}^{\ast } $-algebra of the inverse semigroup associated to $\Lambda $ is the same as the ${C}^{\ast } $-algebra of $\Lambda $.

Keywords

C∗-algebrainverse semigroupcategory of paths

MSC classification

Secondary: 46L05: General theory of $C^*$-algebras 20M18: Inverse semigroups 47D03: Groups and semigroups of linear operators
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 90 , Issue 1 , August 2014 , pp. 121 - 133
Copyright
Copyright ©2014 Australian Mathematical Publishing Association Inc. 

References

Ash, C. J. and Hall, T. E., ‘Inverse semigroups on graphs’, Semigroup Forum 11 (1975/76), 140–145.Google Scholar
Donsig, A., Haataja, S. and Meakin, J., ‘Amalgams of inverse semigroups and ${C}^{\ast } $-algebras’, Indiana Univ. Math. J. 60 (2012), 1059–1076.CrossRefGoogle Scholar
Exel, R., ‘Inverse semigroups and combinatorial ${C}^{\ast } $-algebras’, Bull. Braz. Math. Soc. (N.S.) 39 (2008), 191–313.CrossRefGoogle Scholar
Exel, R., ‘Tight representations of semilattices and inverse semigroups’, Semigroup Forum 79 (2009), 159–182.CrossRefGoogle Scholar
Exel, R., ‘Semigroupoid ${C}^{\ast } $-algebras’, J. Math. Anal. Appl. 377 (2011), 303–318.CrossRefGoogle Scholar
Exel, R., Gonçalves, D. and Starling, C., ‘The tiling ${C}^{\ast } $-algebra viewed as a tight inverse semigroup algebra’, Semigroup Forum 84 (2012), 229–240.Google Scholar
Farthing, C., Muhly, P. and Yeend, T., ‘Higher-rank graph ${C}^{\ast } $-algebras: an inverse semigroup and groupoid approach’, Semigroup Forum 71 (2005), 159–187.Google Scholar
Kellendonk, J., ‘Topological equivalence of tilings’, J. Math. Phys. 38 (1997), 1823–1842.Google Scholar
Kellendonk, J., ‘The local structure of tilings and their integer group of coinvariants’, Comm. Math. Phys. 187 (1997), 115–157.Google Scholar
Koshkam, M. and Skandalis, G., ‘Crossed products of ${C}^{\ast } $-algebras by groupoids and inverse semigroups’, J. Operator Theory 51 (2004), 255–279.Google Scholar
Kumjian, A. and Pask, D., ‘Higher rank graph ${C}^{\ast } $-algebras’, New York J. Math. 6 (2000), 1–20.Google Scholar
Lawson, M. V., Inverse Semigroups: The Theory of Partial Symmetries (World Scientific Publishing Co. Pte. Ltd., Singapore, 1998).CrossRefGoogle Scholar
Lawson, M. V., ‘${E}^{\ast } $-unitary inverse semigroups’, in: Semigroups, Algorithms, Automata, and Languages (Coimbra, 2001) (World Scientific Publishing, River Edge, NJ, 2002), 195–214.CrossRefGoogle Scholar
Lawson, M. V., ‘A noncommutative generalization of Stone duality’, J. Aust. Math. Soc. 88 (2010), 385–404.CrossRefGoogle Scholar
Lawson, M. V., ‘Noncommutative stone duality: inverse semigroups, topological groupoids and ${C}^{\ast } $-algebras’, Internat. J. Algebra Comput. 22 (2012), 1250058.Google Scholar
Lenz, D. H., ‘On an order-based construction of a topological groupoid from an inverse semigroup’, Proc. Edinb. Math. Soc. (2) 51 (2008), 387–406.Google Scholar
Milan, D., ${C}^{\ast } $-Algebras of Inverse Semigroups, PhD Dissertation, Univ. Neb.–Lincoln, May, 2008.Google Scholar
Nica, A., ‘${C}^{\ast } $-algebras generated by isometries and Wiener-Hopf operators’, J. Operator Theory 27 (1992), 17–52.Google Scholar
Paterson, A. L. T., ‘Graph inverse semigroups, groupoids and their ${C}^{\ast } $-algebras’, J. Operator Theory 48 (2002), 645–662.Google Scholar
Spielberg, J., ‘Groupoids and ${C}^{\ast } $-algebras for categories of paths’, Trans. Amer. Math. Soc.(to appear) arXiv:1111.6924.Google Scholar
Steinberg, B., ‘A groupoid approach to discrete inverse semigroup algebras’, Adv. Math. 223 (2010), 689–727.CrossRefGoogle Scholar
Sunder, S., ‘Cuntz–Li relations, inverse semigroups, and groupoids’, Münster J. Math. 5 (2012), 151–182.Google Scholar