C$^{*}$-bialgebra defined by the direct sum of Cuntz-Krieger algebras

Details C$^{*}$-bialgebra defined by the direct sum of Cuntz-Krieger algebras C$^{*}$-bialgebra defined by the direct sum of Cuntz-Krieger algebras

2008-01-30

arxiv.org/abs/0801.4597v2

Mathematics

Operator Algebras

17 pages

Scientific paper

Let ${\sf CK}_{*}$ denote the C$^{*}$-algebra defined by the direct sum of all Cuntz-Krieger algebras. We introduce a comultiplication $\Delta_{\phi}$ and a counit $\epsilon$ on ${\sf CK}_{*}$ such that $\Delta_{\phi}$ is a nondegenerate $*$-homomorphism from ${\sf CK}_{*}$ to ${\sf CK}_{*}\otimes {\sf CK}_{*}$ and $\epsilon$ is a $*$-homomorphism from ${\sf CK}_{*}$ to ${\bf C}$. From this, ${\sf CK}_{*}$ is a counital non-commutative non-cocommutative C$^{*}$-bialgebra. Furthermore, C$^{*}$-bialgebra automorphisms, a tensor product of representations and C$^{*}$-subbialgebras of ${\sf CK}_{*}$ are investigated.

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