Semigroupoid C⁎-algebras

A semigroupoid is a set equipped with a partially defined associative operation. Given a semigroupoid Λ we construct a C⁎-algebra O(Λ) from it. We then present two main examples of semigroupoids, namely the Markov semigroupoid associated to an infinite 0-1 matrix, and the semigroupoid associated to a row-finite higher-rank graph without sources. In both cases the semigroupoid C⁎-algebra is shown to be isomorphic to the algebras usually attached to the corresponding combinatorial object, namely the Cuntz-Krieger algebras and the higher-rank graph C⁎-algebras, respectively. In the case of a higher-rank graph (Λ,d), it follows that the dimension function d is superfluous for defining the corresponding C⁎-algebra.