On nuclearity of the C∗-algebra of an inverse semigroup

• Autores: Massoud Amini, Mahmood Khoshkam
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 64, Nº 2, 2020, págs. 499-511
• Idioma: inglés
• DOI: 10.5565/publmat6422005
• Enlaces
  • Texto completo
• Resumen

  • 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∗(ES) on the idempotents ES of S are amenable. Thereby we solve some problems raised by A. L. T. Paterson.

• Referencias bibliográficas
  • C. Anantharaman-Delaroche and J. Renault, “Amenable Groupoids”, With a foreword by G. Skandalis and Appendix B by E. Germain, Monographies...
  • C. Berg, J. P. R. Christensen, and P. Ressel, “Harmonic Analysis on Semigroups. Theory of Positive Definite and Related Functions”, Graduate...
  • M.-D. Choi and E. G. Effros, Nuclear C∗-algebras and injectivity: the general case, Indiana Univ. Math. J. 26(3) (1977), 443–446. DOI: 10.1512/iumj.1977....
  • A. Connes, On the cohomology of operator algebras, J. Functional Analysis 28(2) (1978), 248–253. DOI: 10.1016/0022-1236(78)90088-5.
  • A. Connes, J. Feldman, and B. Weiss, An amenable equivalence relation is generated by a single transformation, Ergodic Theory Dynam. Systems...
  • J. Duncan and I. Namioka, Amenability of inverse semigroups and their semigroup algebras, Proc. Roy. Soc. Edinburgh Sect. A 80(3–4) (1978),...
  • R. Exel and C. Starling, Amenable actions of inverse semigroups, Ergodic Theory Dynam. Systems 37(2) (2017), 481–489. DOI: 10.1017/etds.2015.60.
  • U. Haagerup, All nuclear C∗-algebras are amenable, Invent. Math. 74(2) (1983), 305–319. DOI: 10.1007/BF01394319.
  • B. E. Johnson, “Cohomology in Banach Algebras”, Memoirs of the American Mathematical Society 127, American Mathematical Society, Providence,...
  • M. Khoshkam and G. Skandalis, Regular representation of groupoid C∗-algebras and applications to inverse semigroups, J. Reine Angew. Math....
  • M. Khoshkam and G. Skandalis, Crossed products of C∗-algebras by groupoids and inverse semigroups, J. Operator Theory 51(2) (2004), 255–279.
  • D. Milan, C∗-algebras of inverse semigroups: amenability and weak containment, J. Operator Theory 63(2) (2010), 317–332.
  • A. L. T. Paterson, “Amenability”, Mathematical Surveys and Monographs 29, American Mathematical Society, Providence, RI, 1988. DOI: 10.1090/surv/029.
  • A. L. T. Paterson, “Groupoids, Inverse Semigroups, and their Operator Algebras”, Progress in Mathematics 170, Birkh¨auser Boston, Inc., Boston,...
  • A. L. T. Paterson, Graph inverse semigroups, groupoids and their C∗-algebras, J. Operator Theory 48(3), suppl. (2002), 645–662.
  • M. Petrich, “Inverse Semigroups”, Pure and Applied Mathematics (New York), A Wiley-Interscience Publication. John Wiley & Sons, Inc.,...
  • J. Renault, “A Groupoid Approach to C∗-Algebras”, Lecture Notes in Mathematics 793, Springer, Berlin, 1980. DOI: 10.1007/BFb0091072.
  • B. Steinberg, A groupoid approach to discrete inverse semigroup algebras, Adv. Math. 223(2) (2010), 689–727. DOI: 10.1016/j.aim.2009.09.001.