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’s type characterizations of amenability and give an example of a semigroup whose semigroup ring is algebraically amenable but has no Føln...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión enviada para evaluación y publicación |
| Fecha de publicación: | 2019 |
| País: | España |
| Institución: | Consejo Superior de Investigaciones Científicas (CSIC) |
| Repositorio: | DIGITAL.CSIC. Repositorio Institucional del CSIC |
| OAI Identifier: | oai:digital.csic.es:10261/197106 |
| Acceso en línea: | http://hdl.handle.net/10261/197106 |
| Access Level: | acceso abierto |
| Palabra clave: | Amenability Paradoxical decompositions Følner condition Semigroups Semigroup rings Inverse semigroup C*-algebra Proper infiniteness Amenable traces |
| Sumario: | We analyze the dichotomy amenable/paradoxical in the context of (discrete, countable,unital) semigroups and corresponding semigroup rings. We consider also Følner’s 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 RX. 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 RX; (5) RX has an amenable trace; (6) RX is not properly infinite and (7) [0] 6= [1] in the K0-group of RX . We also show that any tracial state on RX 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 ∗ (X) implies the amenability of X. The converse implication is false. |
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