On the inverse semigroup of bimodules over a C*-algebra

TL;DR

This paper generalizes a construction of an inverse semigroup from metric spaces to C*-algebras, specifically using Hilbert bimodules, and explores its properties and relations to coarse geometry.

Contribution

It introduces an inverse semigroup of Hilbert bimodules over a C*-algebra, extending the metric space construction to a broader algebraic context.

Findings

Constructed an inverse semigroup $S(A)$ of bimodules for a C*-algebra $A$.

Established an injective map from the inverse semigroup of metrics to $S(C^*_u(X))$.

Showed the map is not surjective, leading to a metric-independent inverse semigroup.

Abstract

It was noticed recently that, given a metric space $(X,d_X)$, the equivalence classes of metrics on the disjoint union of the two copies of $X$ coinciding with $d_X$ on each copy form an inverse semigroup $M(X)$ with respect to concatenation of metrics. Now put this inverse semigroup construction in a more general context, namely, we define, for a C*-algebra $A$, an inverse semigroup $S(A)$ of Hilbert C*-$A$-$A$-bimodules. When $A$ is the uniform Roe algebra $C^*_u(X)$ of a metric space $X$, we construct a map $M(X)\to S(C^*_u(X))$ and show that this map is injective, but not surjective in general. This allows to define an analog of the inverse semigroup $M(X)$ that does not depend on the choice of a metric on $X$ within its coarse equivalence class.