Abstract
The direct sum \({{\mathcal O}_{*}}\) of all Cuntz algebras has a non-cocommutative comultiplication \({\Delta_{\varphi}}\) such that there exists no antipode of any dense subbialgebra of the C*-bialgebra \({({\mathcal O}_{*},\Delta_{\varphi})}\). From states equations of \({{\mathcal O}_{*}}\) with respect to the tensor product, we construct an operator W for \({({\mathcal O}_{*},\Delta_{\varphi})}\) such that W* is an isometry, \({W(x\otimes I)W^{*}=\Delta_{\varphi}(x)}\) for each \({x\in {\mathcal O}_{*}}\) and W satisfies the pentagon equation.
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Kawamura, K. Pentagon Equation Arising from State Equations of a C*-Bialgebra. Lett Math Phys 93, 229–241 (2010). https://doi.org/10.1007/s11005-010-0413-5
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DOI: https://doi.org/10.1007/s11005-010-0413-5