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

15 pages