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

Abstract

We show that a tensor product among representation of certain C$^{*}$-algebras induces a bialgebra. Let $\tilde{\cal O}_{*}$ be the smallest unitization of the direct sum of Cuntz algebras \[{\cal O}_{*}\equiv {\bf C}\oplus {\cal O}_{2}\oplus {\cal O}_{3}\oplus{\cal O}_{4}\oplus ....\] We show that there exists a non-cocommutative comultiplication $Δ$ and a counit $ε$ of $\tilde{\cal O}_{*}$. From $Δ,\vep$ and the standard algebraic structure, $\tilde{\cal O}_{*}$ is a C$^{*}$-bialgebra. Furthermore we show the following: (i) The antipode on $\tilde{\cal O}_{*}$ never exist. (ii) There exists a unique Haar state on $\tilde{\cal O}_{*}$. (iii) For a certain one-parameter bialgebra automorphism group of $\tilde{\cal O}_{*}$, a KMS state on $\tilde{\cal O}_{*}$ exists.