Semigroupoid -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.