  Let ${\cal A}_{0}(*)$ denote the direct sum of a certain set of UHF algebras
and let ${\cal A}(*)\equiv {\bf C}\oplus {\cal A}_{0}(*)$. We introduce a
non-cocommutative comultiplication $\Delta_{\phi}$ on ${\cal A}(*)$, and give
an example of comodule-C$^{*}$-algebra of the C$^{*}$-bialgebra $({\cal
A}(*),\Delta_{\phi})$. With respect to $\Delta_{\phi}$, we define a
non-symmetric tensor product of *-representations of UHF algebras and show
tensor product formulas of GNS representations by product states.
