Pentagon equation arising from state equations of a C$^{*}$-bialgebra

Details Pentagon equation arising from state equations of a C$^{*}$-bialgebra Pentagon equation arising from state equations of a C$^{*}$-bialgebra

2009-06-14

arxiv.org/abs/0906.2507v1

Mathematics

Operator Algebras

15 pages

Scientific paper

The direct sum ${\cal O}_{*}$ 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 $({\cal O}_{*},\Delta_{\varphi})$. From states equations of ${\cal O}_{*}$ with respect to the tensor product, we construct an operator $W$ for $({\cal O}_{*},\Delta_{\varphi})$ such that $W^{*}$ is an isometry, $W(x\otimes I)W^{*}=\Delta_{\varphi}(x)$ for each $x\in {\cal O}_{*}$ and $W$ satisfies the pentagon equation.

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