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 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]≠[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 Cr⁎(X) implies the amenability of X. The reverse implication (which is a natural generalization of Rosenberg's conjecture to this context) is false.