  We consider the $C^*$-algebra generated by finitely many annihilation
operators acting on the weakly monotone Fock space, and we call it weakly
monotone $C^*$-algebra. We give an abstract representation for this algebra,
showing that it is isomorphic to a suitable quotient of a Cuntz-Krieger
$C^*$-algebra $\mathcal{O}_A$ corresponding to a suitable matrix $A$.
Furthermore, we show that the diagonal subalgebra of the weakly monotone
$C^*$-algebra is a MASA and we give the detailed description of its Gelfand
spectrum.
