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

Abstract

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