Biideals and a lattice of C${}^{\ast }$-bialgebras associated with prime numbers

Let ${\mathcal{O}}_{\ast }$ be the C${}^{\ast }$-algebra defined as the direct sum of all Cuntz algebras. Then ${\mathcal{O}}_{\ast }$ has a non-cocommutative comultiplication ${\mathrm{\Delta }}_{\varphi }$ and a counit $ϵ$. Let $\mathrm{B}\mathrm{I}({\mathcal{O}}_{\ast })$ denote the set of all closed biideals of the C${}^{\ast }$-bialgebra $({\cal O}{*},\Delta{\phi},\epsilon)$andlet${\cal P}({\bf P})$ denote the power set of the set of all prime numbers. We show a one-to-one correspondence between $\mathrm{B}\mathrm{I}({\mathcal{O}}_{\ast })$ and $\mathcal{P}(\mathbf{P})$. Furthermore, we show that for any $\mathcal{I}$ in $\mathrm{B}\mathrm{I}({\mathcal{O}}_{\ast })$, there exists a C${}^{\ast }$-subbialgebra ${\mathcal{B}}_{\mathcal{I}}$ of ${\mathcal{O}}_{\ast }$ such that ${\cal O}{*}={\cal B}{{\cal I}}\oplus {\cal I}$, and the set of all such C${}^{\ast }$-subbialgebras is a lattice with respect to the natural operations among C${}^{\ast }$-subbialgebras, which is isomorphic to the lattice $\mathcal{P}(\mathbf{P})$.