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

2009/04/28 by Kawamura, Katsunori
#06B05 #11A41 #16W30 #FOS: Mathematics #Operator Algebras (math.OA) #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.0904.4296

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 Δ_ϕ and a counit ε. Let \rm BI(\cal O_*) denote the set of all closed biideals of the C^*-bialgebra (\cal O_*,Δ_ϕ,ε) 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}⊕ \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).

Related

• C^*-bialgebra defined by the direct sum of Cuntz-Krieger algebras
• C^*-bialgebra defined by the direct sum of Cuntz algebras
• C^*-bialgebra defined as the direct sum of UHF algebras
• Non-cocommutative C^*-bialgebra defined as the direct sum of free group C^*-algebras
• Subquotients of Hecke C^*-algebras