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

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dc.creatorKawamura, Katsunori
dc.date2007-02-13
dc.date2007-03-14
dc.date.accessioned2026-07-07T07:51:41Z
dc.date.available2026-07-07T07:51:41Z
dc.descriptionWe 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.
dc.description18 pages
dc.identifierhttps://arxiv.org/abs/math/0702355
dc.identifierhttp://arxiv.org/abs/math/0702355
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/125602
dc.subjectOperator Algebras
dc.subject47L55; 81T05
dc.titleC$^{*}$-bialgebra defined by the direct sum of Cuntz algebras
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