  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 (quotient 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 $I$.
