Amenability and paradoxicality in semigroups and C*-algebras

Abstract

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 ${\mathcal{R}}_X$. 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 ${\mathcal{R}}_X$; (5) ${\mathcal{R}}_X$ has an amenable trace; (6) ${\mathcal{R}}_X$ is not properly infinite and (7) $[0]\ne [1]$ in the $K_0$-group of ${\mathcal{R}}_X$. We also show that any tracial state on ${\mathcal{R}}_X$ 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 $C_r^{\ast }\left(X\right)$ implies the amenability of $X$. The converse implication is false.

arXiv comment29 pages, minor corrections. Mistake in the statement of Proposition 4.19 from previous version corrected. Final version to appear in Journal of Functional Analysis

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Cited by in corpus (2)

• The uniform Roe algebra of an inverse semigroup
• A note on the quasi-diagonality of inverse semigroup reduced C*-algebras