Discrete Conduché fibrations and \(C^\ast\)-algebras (Q2362665)

A \(k\)-graph is a fibration \(d : \Lambda \to \mathbb{N}^k\) such that every factorization of \(d(\lambda)\) lifts uniquely to a factorization of \(\lambda\). Such \(k\)-graphs induce very interesting \(C^\ast\)-algebras. They are also instances of discrete Conduché fibrations: fibrations \(F\) such that every factorization of \(F(f)\) lifts uniquely to a factorization of \(f\). The authors generalize the theory of \(k\)-graph \(C^\ast\)-algebras to \(C^\ast\)-algebras induced by discrete Conduché fibrations. First, it is shown that the Cuntz-Kreiger relations on a discrete Conduché fibration allow a universal \(C^\ast\)-algebra. Their theory includes \(C^\ast\)-algebras induced by categories of paths as follows. If the base category is strong right Ore, infinite paths may be regarded as sections of slices of the fibration, and behave similarly as for \(k\)-graph \(C^\ast\)-algebras. In particular, it follows that discrete Conduché fibration \(C^\ast\)-algebras are nontrivial. Topologizing the set of infinite paths gives a locally compact Hausdorff étale groupoid. If the base category is left-cancellative, the resulting groupoid \(C^\ast\)-algebra coincides with the discrete Conduché fibration \(C^\ast\)-algebra. If the base category is moreover right-cancellative, then the total category embeds into operators on a Hilbert space.