  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.
