C${}^{\ast }$-bialgebra defined as the direct sum of UHF algebras

Let ${\mathcal{A}}_0(\ast )$ denote the direct sum of a certain set of UHF algebras and let $\mathcal{A}(\ast )\equiv \mathbf{C}\oplus {\mathcal{A}}_0(\ast )$. We introduce a non-cocommutative comultiplication ${\mathrm{\Delta }}_{\varphi }$ on $\mathcal{A}(\ast )$, and give an example of comodule-C${}^{\ast }$-algebra of the C${}^{\ast }$-bialgebra $({\cal A}(*),\Delta_{\phi})$\mathrm{.}Withrespectto$\Delta_{\phi}$, we define a non-symmetric tensor product of *-representations of UHF algebras and show tensor product formulas of GNS representations by product states.