Finitely generated weakly monotone C*-algebra

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