Inverse semigroups of $C^*$-algebras and metric spaces

The talk is based on the observation that Hilbert bimodules over a $C^*$-algebra have a natural structure of an inverse semigroup. In the case when the $C^*$-algebra is a Roe algebra of a metric space $X$, a relation of this semigroup to the semigroup of metrics on the double of $X$ considered earlier will be shown. For the latter, new results will be given, showing, in particular, its dependence from a metric in the equivalence class. Also presence or absence of non-unitary isometries and its relation to geometry of $X$ will be discussed.