Inverse semigroup from metrics on doubles, III: Commutativity and (in)finiteness of idempotents

We have shown recently that, given a metric space X, the coarse equivalence classes of metrics on the two copies of X form an inverse semigroup M(X).Here we study the property of idempotents in M(X) of being finite or infinite, which is similar to this property for projections in C * -algebras.We show that if X is a free group then the unit of M(X) is infinite, while if X is a free abelian group then it is finite.As a by-product, we show that the inverse semigroup M(X) is not a quasiisometry invariant.We also show that M(X) is commutative if it is Clifford, and give a geometric description of spaces X for which M(X) is commutative.• The restriction of d onto X and Y are d X and d Y respectively.• inf x∈X,y∈Y d(x, y) > 0.