×

On nuclearity of the \(C^*\)-algebra of an inverse semigroup. (English) Zbl 1460.46049

Summary: We show that the universal groupoid of an inverse semigroup \(S\) is topologically (measurewise) amenable if and only if \(S\) is hyperfinite and all members of a family of subsemigroups of \(S\) indexed by the spectrum of the commutative \(C^*\)-algebra \(C^*(E_S)\) on the idempotents \(E_S\) of \(S\) are amenable. Thereby we solve some problems raised by A. L. T. Paterson [Groupoids, inverse semigroups, and their operator algebras. Boston, MA: Birkhäuser (1999; Zbl 0913.22001)].

MSC:

46L35 Classifications of \(C^*\)-algebras
46L57 Derivations, dissipations and positive semigroups in \(C^*\)-algebras
43A07 Means on groups, semigroups, etc.; amenable groups

Citations:

Zbl 0913.22001
PDFBibTeX XMLCite
Full Text: DOI Euclid

References:

[1] C. Anantharaman-Delaroche and J. Renault,“Amenable Groupoids”, With a foreword by G. Skandalis and Appendix B by E. Germain, Monographies de L’Enseignement Math´ematique36, L’Enseignement Math´ematique, Geneva, 2000. · Zbl 0960.43003
[2] C. Berg, J. P. R. Christensen, and P. Ressel,“Harmonic Analysis on Semigroups. Theory of Positive Definite and Related Functions”, Graduate Texts in Mathematics100, Springer-Verlag, New York, 1984.DOI: 10.1007/ 978-1-4612-1128-0.
[3] M.-D. Choi and E. G. Effros, NuclearC∗-algebras and injectivity: the general case,Indiana Univ. Math. J.26(3)(1977), 443-446.DOI: 10.1512/iumj.1977. 26.26034. · Zbl 0378.46052
[4] A. Connes, On the cohomology of operator algebras,J. Functional Analysis 28(2)(1978), 248-253.DOI: 10.1016/0022-1236(78)90088-5. · Zbl 0408.46042
[5] A. Connes, J. Feldman, and B. Weiss, An amenable equivalence relation is generated by a single transformation,Ergodic Theory Dynam. Systems1(4) (1981), 431-450 (1982).DOI: 10.1017/s014338570000136x. · Zbl 0491.28018
[6] J. Duncan and I. Namioka, Amenability of inverse semigroups and their semigroup algebras,Proc. Roy. Soc. Edinburgh Sect. A80(3-4)(1978), 309-321. DOI: 10.1017/S0308210500010313. · Zbl 0393.22004
[7] R. Exel and C. Starling, Amenable actions of inverse semigroups,Ergodic Theory Dynam. Systems37(2)(2017), 481-489.DOI: 10.1017/etds.2015.60. · Zbl 1388.46047
[8] U. Haagerup, All nuclearC∗-algebras are amenable,Invent. Math.74(2) (1983), 305-319.DOI: 10.1007/BF01394319. · Zbl 0529.46041
[9] B. E. Johnson,“Cohomology in Banach Algebras”, Memoirs of the American Mathematical Society127, American Mathematical Society, Providence, R.I., 1972. · Zbl 0256.18014
[10] M. Khoshkam and G. Skandalis, Regular representation of groupoidC∗-algebras and applications to inverse semigroups,J. Reine Angew. Math.2002(546) (2002), 47-72.DOI: 10.1515/crll.2002.045. · Zbl 1029.46082
[11] M. Khoshkam and G. Skandalis, Crossed products ofC∗-algebras by groupoids and inverse semigroups,J. Operator Theory51(2)(2004), 255-279. · Zbl 1061.46047
[12] D. Milan,C∗-algebras of inverse semigroups: amenability and weak containment,J. Operator Theory63(2)(2010), 317-332. · Zbl 1212.46079
[13] A. L. T. Paterson,“Amenability”, Mathematical Surveys and Monographs29, American Mathematical Society, Providence, RI, 1988.DOI: 10.1090/surv/029.
[14] A. L. T. Paterson,“Groupoids, Inverse Semigroups, and their Operator Algebras”, Progress in Mathematics170, Birkh¨auser Boston, Inc., Boston, MA, 1999.DOI: 10.1007/978-1-4612-1774-9. · Zbl 0913.22001
[15] A. L. T. Paterson, Graph inverse semigroups, groupoids and theirC∗-algebras, J. Operator Theory48(3), suppl. (2002), 645-662. · Zbl 1031.46060
[16] M. Petrich,“Inverse Semigroups”, Pure and Applied Mathematics (New York), A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1984. · Zbl 0546.20053
[17] J. Renault,“A Groupoid Approach toC∗-Algebras”, Lecture Notes in Mathematics793, Springer, Berlin, 1980.DOI: 10.1007/BFb0091072. · Zbl 0433.46049
[18] B.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.