  We show that a tensor product among representation of certain
C$^{*}$-algebras induces a bialgebra. Let $\tilde{{\cal O}}_{*}$ be the
smallest unitization of the direct sum of Cuntz algebras \[{\cal O}_{*}\equiv
{\bf C}\oplus {\cal O}_{2}\oplus {\cal O}_{3}\oplus{\cal O}_{4}\oplus ....\] We
show that there exists a non-cocommutative comultiplication $\Delta$ and a
counit $\epsilon$ of $\tilde{{\cal O}}_{*}$. From $\Delta,\vep$ and the
standard algebraic structure, $\tilde{{\cal O}}_{*}$ is a C$^{*}$-bialgebra.
Furthermore we show the following: (i) The antipode on $\tilde{{\cal O}}_{*}$
never exist. (ii) There exists a unique Haar state on $\tilde{{\cal O}}_{*}$.
(iii) For a certain one-parameter bialgebra automorphism group of $\tilde{{\cal
O}}_{*}$, a KMS state on $\tilde{{\cal O}}_{*}$ exists.
