-bialgebras associated with prime numbers

Let O∗ be the C-algebra defined as the direct sum of all Cuntz algebras. Then O∗ has a non-cocommutative comultiplication ∆φ and a counit ε. Let BI(O∗) denote the set of all closed biideals of the C-bialgebra (O∗,∆φ, ε) and let P(P) denote the power set of the set of all prime numbers. We show a one-to-one correspondence between BI(O∗) and P(P). Furthermore, we show that for any I in BI(O∗), there exists a C-subbialgebra BI ofO∗ such that O∗ = BI⊕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 P(P). Mathematics Subject Classifications (2000). 16W30, 06B05, 11A41.