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