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

An abstract characterization of weakly monotone \(C^*\) $C^{\ast }$ -algebras, namely the concrete \(C^*\) $C^{\ast }$ -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 \(C^*\) $C^{\ast }$ -algebra indexed by \({\mathbb N}\) $\mathbb{N}$ is shown to be a type-I \(C^*\) $C^{\ast }$ -algebra and its representation theory is entirely determined, whereas the weakly monotone \(C^*\) $C^{\ast }$ -algebra indexed by \({\mathbb Z}\) $\mathbb{Z}$ is shown not to be of type I.