Pentagon equation arising from state equations of a C${}^{\ast }$-bialgebra

The direct sum ${\mathcal{O}}_{\ast }$ of all Cuntz algebras has a non-cocommutative comultiplication ${\mathrm{\Delta }}_{\phi }$ such that there exists no antipode of any dense subbialgebra of the C${}^{\ast }$-bialgebra $({\mathcal{O}}_{\ast },{\mathrm{\Delta }}_{\phi })$. From states equations of ${\mathcal{O}}_{\ast }$ with respect to the tensor product, we construct an operator $W$ for $({\mathcal{O}}_{\ast },{\mathrm{\Delta }}_{\phi })$ such that $W^{\ast }$ is an isometry, $W(x\otimes I)W^{\ast }={\mathrm{\Delta }}_{\phi }(x)$ for each $x\in {\cal O}_{*}$and$W$ satisfies the pentagon equation.