On the Inverse Semigroup of Bimodules over a \(C^*\)-Algebra

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 mapping \(M(X)\to S(C^*_u(X))\) and show that this mapping 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.