C*-algebras from partial isometric representations of LCM semigroups

Abstract

We give a new construction of a C*-algebra from a cancellative semigroup $P$ via partial isometric representations, generalising 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.

arXiv comment38 pages. The title has changed; previously this was called "The C*-algebra of a cancellative semigroup." The introduction has been significantly expanded, with some added motivation and updated references. The proof of the last result has changed between v3 and v4 (though both are correct). Version 4 matches the version published in Münster J. Math