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.

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

References in corpus (1)

• $C^{\ast }$-algebras of inverse semigroups: amenability and weak containment

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