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

Abstract

We show that a tensor product among representation of certain C${}^{\ast }$-algebras induces a bialgebra. Let ${\tilde{\mathcal{O}}}_{\ast }$ 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 $\Delta$ and a counit $\epsilon$ of ${\tilde{\mathcal{O}}}_{\ast }$. From $Δ,\vep$ and the standard algebraic structure, ${\tilde{\mathcal{O}}}_{\ast }$ is a C${}^{\ast }$-bialgebra. Furthermore we show the following: (i) The antipode on ${\tilde{\mathcal{O}}}_{\ast }$ never exist. (ii) There exists a unique Haar state on ${\tilde{\mathcal{O}}}_{\ast }$. (iii) For a certain one-parameter bialgebra automorphism group of ${\tilde{\mathcal{O}}}_{\ast }$, a KMS state on ${\tilde{\mathcal{O}}}_{\ast }$ exists.

18 pages

References in corpus (1)

• A tensor product of representations of Cuntz algebras