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 higher-rank graph. 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. We moreover present a proposal for extending the construction of C * (Λ) to the case when Λ is not row-finite or contains sources.