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C * -algebras of inverse semigroups: amenability and weak containment. (English) Zbl 1212.46079
In the last four decades equivalent definitions of group amenability in various contexts were found. Not all of the equivalent definitions of group amenability translate well to inverse semigroups. The present paper is dedicated to this problem, claiming that the weak containment property, motivated by group theory, is an appropriate notion of amenability for inverse semigroups. Throughout the six sections of the paper, starting with an introduction and a preliminary section, where all necessary definitions are introduced, the obtained results suggest that weak containment is the right notion of amenability for inverse semigroups. It is pointed out that a strong E * -unitary inverse semigroup S has weak containment if and only if the associated Fell bundle over the universal group is amenable and that the graph inverse semigroups have weak containment. Also, various other situations are analysed. In the third section, related results for inverse semigroups with zero and weak containment property are analysed. In the fourth section it is shown that all graph inverse semigroups have weak containment, yet the universal grading of a graph inverse semigroup is a free group. The fifth section is dedicated to Nica’s inverse semigroup 𝒯 G,P [A. Nica, J. Oper. Theory 27, No. 1, 17–52 (1992; Zbl 0809.46058)] induced from a quasi-lattice ordered group (G,P), and it is pointed out that Nica’s definition of amenability of a quasi-lattice ordered group (G,P) is equivalent to weak containment for 𝒯 G,P . In the last section of the paper various properties concerning the positivity are analysed, examples are given, and some open questions are raised.
MSC:
46L05General theory of C * -algebras
20M18Inverse semigroups