C^*-bialgebra defined by the direct sum of Cuntz algebras

2007/02/13 by Katsunori Kawamura, Kawamura, Katsunori
Mathematics · #47L55 #81T05 #FOS: Mathematics #Operator Algebras (math.OA) #math.OA #msc:47L55 #msc:81T05

paper · pdf · doi:10.48550/arxiv.math/0702355

18 pages

arxiv created 2007/03/14 · arxiv updated 2009/12/01

Abstract

We show that a tensor product among representation of certain C^*-algebras induces a bialgebra. Let \cal O_* be the smallest unitization of the direct sum of Cuntz algebras \cal O_*≡ \bf C⊕ \cal O₂⊕ \cal O₃⊕\cal O₄⊕ .... We show that there exists a non-cocommutative comultiplication Δ and a counit ε of \cal O_*. From Δ,\vep and the standard algebraic structure, \cal O_* is a C^*-bialgebra. Furthermore we show the following: (i) The antipode on \cal O_* never exist. (ii) There exists a unique Haar state on \cal O_*. (iii) For a certain one-parameter bialgebra automorphism group of \cal O_*, a KMS state on \cal O_* exists.

Related

• C^*-bialgebra defined by the direct sum of Cuntz-Krieger algebras
• C^*-bialgebra defined as the direct sum of UHF algebras
• Biideals and a lattice of C^*-bialgebras associated with prime numbers
• Non-cocommutative C^*-bialgebra defined as the direct sum of free group C^*-algebras
• The Cuntz Algebra Q_N and C*-Algebras of Product Systems