Weakly-monotone \(C^*\)-algebras as Exel–Laca algebras

Abstract

An abstract characterization of weakly monotone \(C^*\)-algebras, namely the concrete \(C^*\)-algebras generated by creators and annihilators acting on the so-called weakly monotone Fock spaces, is given in terms of (quotients of) suitable Exel–Laca algebras. The weakly monotone \(C^*\)-algebra indexed by \({\mathbb N}\) is shown to be a type-I \(C^*\)-algebra and its representation theory is entirely determined, whereas the weakly monotone \(C^*\)-algebra indexed by \({\mathbb Z}\) is shown not to be of type I.