C*-algebras from partial isometric representations of LCM semigroups

We give a new construction of a C*-algebra from a cancellative semigroup P via partial isometric representations, generalizing the construction from the second named author’s thesis. We then study our construction in detail for the special case when P is an LCM semigroup. In this case, we realize our algebras as inverse semigroup algebras and groupoid algebras, and apply our construction to free semigroups and Zappa–Szép products associated to self-similar groups