Amenability and paradoxicality in semigroups and C*-algebras
We analyze the dichotomy amenable/paradoxical in the context of (discrete, countable, unital) semigroups and corresponding semigroup rings. We consider also Følner type characterizations of amenability and give an example of a semigroup whose semigroup ring is algebraically amenable but has no Følne...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2020 |
| País: | España |
| Institución: | Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
| Repositorio: | Recercat. Dipósit de la Recerca de Catalunya |
| OAI Identifier: | oai:recercat.cat:2072/530623 |
| Acceso en línea: | http://hdl.handle.net/2072/530623 |
| Access Level: | acceso abierto |
| Palabra clave: | Matemàtiques 51 |
| Sumario: | We analyze the dichotomy amenable/paradoxical in the context of (discrete, countable, unital) semigroups and corresponding semigroup rings. We consider also Følner type characterizations of amenability and give an example of a semigroup whose semigroup ring is algebraically amenable but has no Følner sequence. In the context of inverse semigroups S we give a characterization of invariant measures on S (in the sense of Day) in terms of two notions: domain measurability and localization. Given a unital representation of S in terms of partial bijections on some set X we define a natural generalization of the uniform Roe algebra of a group, which we denote by R X . We show that the following notions are then equivalent: (1) X is domain measurable; (2) X is not paradoxical; (3) X satisfies the domain Følner condition; (4) there is an algebraically amenable dense*-subalgebra of R X ; (5) R X has an amenable trace; (6) R X is not properly infinite and (7) [0] = [1] in the K 0 -group of R X . We also show that any tracial state on R X is amenable. Moreover, taking into account the localization condition, we give several C*-algebraic characterizations of the amenability of X. Finally, we show that for a certain class of inverse semigroups, the quasidiagonality of C r ∗ (X) implies the amenability of X. The reverse implication (which is a natural generalization of Rosenberg’s conjecture to this context)is false. |
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