Weakly-monotone C*-algebras as Exel-Laca algebras

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