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

Abstract

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

17 pages

Cited by in corpus (1)

• Tensor products of type III factor representations of Cuntz-Krieger algebras