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

Let ${\mathsf{C}\mathsf{K}}_{\ast }$ denote the C${}^{\ast }$-algebra defined by the direct sum of all Cuntz-Krieger algebras. We introduce a comultiplication ${\mathrm{\Delta }}_{\varphi }$ and a counit $ϵ$ on ${\mathsf{C}\mathsf{K}}_{\ast }$ such that ${\mathrm{\Delta }}_{\varphi }$ is a nondegenerate $\ast$-homomorphism from ${\mathsf{C}\mathsf{K}}_{\ast }$ to ${\sf CK}{*}\otimes {\sf CK}{}$and$\epsilon$isa$$-homomorphismfrom${\sf CK}_{*}$to${\bf C}$. From this, ${\mathsf{C}\mathsf{K}}_{\ast }$ 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 ${\mathsf{C}\mathsf{K}}_{\ast }$ are investigated.