Non-cocommutative C$^{*}$-bialgebra defined as the direct sum of free group C$^{*}$-algebras

Let ${\Bbb F}_{n}$ be the free group of rank $n$ and let $\bigoplus C^{*}({\Bbb F}_{n})$ denote the direct sum of full group C$^{*}$-algebras $C^{*}({\Bbb F}_{n})$ of ${\Bbb F}_{n}$ $(1\leq n<\infty$). We introduce a new comultiplication $\Delta_{\varphi}$ on $\bigoplus C^{*}({\Bbb F}_{n})$ such that $(\bigoplus C^{*}({\Bbb F}_{n}),\,\Delta_{\varphi})$ is a non-cocommutative C$^{*}$-bialgebra. With respect to $\Delta_{\varphi}$, the tensor product $\pi\otimes_{\varphi}\pi'$ of any two representations $\pi$ and $\pi'$ of free groups is defined. The operation $\ptimes$ is associative and non-commutative. We compute its tensor product formulas of several representations.