The uniform Roe algebra of an inverse semigroup

Abstract

Given a discrete and countable inverse semigroup $S$ one can study, in analogy to the group case, its geometric aspects. In particular, we can equip $S$ with a natural metric, given by the path metric in the disjoint union of its Schützenberger graphs. This graph, which we denote by ${\Lambda }_S$, inherits much of the structure of $S$. In this article we compare the C*-algebra ${\mathcal{R}}_S$, generated by the left regular representation of $S$ on ${\ell }^2(S)$ and ${\ell }^{\infty }(S)$, with the uniform Roe algebra over the metric space, namely $C_u^{\ast }({\Lambda }_S)$. This yields a chacterization of when ${\mathcal{R}}_S=C_u^{\ast }({\Lambda }_S)$, which generalizes finite generation of $S$. We have termed this by finite labeability (FL), since it holds when the ${\Lambda }_S$ can be labeled in a finitary manner. The graph ${\Lambda }_S$, and the FL condition above, also allow to analyze large scale properties of ${\Lambda }_S$ and relate them with C*-properties of the uniform Roe algebra. In particular, we show that domain measurability of $S$ (a notion generalizing Day's definition of amenability of a semigroup, cf., [5]) is a quasi-isometric invariant of ${\Lambda }_S$. Moreover, we characterize property A of ${\Lambda }_S$ (or of its components) in terms of the nuclearity and exactness of the corresponding C*-algebras. We also treat the special classes of F-inverse and E-unitary inverse semigroups from this large scale point of view.

arXiv commentAccepted version for publication; minor changes

References in corpus (14)

• Amenable actions and exactness for discrete groups
• Some notes on property A
• Amenability and uniform Roe algebras
• A non-amenable groupoid whose maximal and reduced $C^{\ast }$-algebras are the same
• Some remarks about the weak containment property for groupoids and semigroups
• Amenability of coarse spaces and K-algebras
• $C^{\ast }$-algebras of inverse semigroups: amenability and weak containment
• C*-crossed products by partial actions and actions of inverse semigroups
• Amenable actions of inverse semigroups
• Groupoid exactness and the weak containment problem
• Inverse monoids and immersions of 2-complexes
• Amenability and paradoxicality in semigroups and C*-algebras
• Finite-dimensional approximation properties for uniform Roe algebras
• General uniform Roe algebra rigidity

Cited by in corpus (2)

• A note on the quasi-diagonality of inverse semigroup reduced C*-algebras
• Geometric Inverse Semigroup Theory: a note on the Milnor-Schwarz Lemma for inverse monoids