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

TL;DR

This paper constructs a new C$^{*}$-bialgebra from the direct sum of all Cuntz-Krieger algebras, introducing comultiplication and counit, and explores its automorphisms, representations, and subbialgebras.

Contribution

It defines a novel C$^{*}$-bialgebra structure on the direct sum of Cuntz-Krieger algebras and investigates its algebraic properties and symmetries.

Findings

${ m CK}_*$ is a counital non-commutative non-cocommutative C$^{*}$-bialgebra.

Automorphisms and subbialgebras of ${ m CK}_*$ are characterized.

Tensor products of representations of ${ m CK}_*$ are studied.

Abstract

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.