%0 Journal Article
%T Pentagon equation arising from state equations of a C$^{*}$-bialgebra
%A Katsunori Kawamura
%J Mathematics
%D 2009
%I arXiv
%R 10.1007/s11005-010-0413-5
%X 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.
%U http://arxiv.org/abs/0906.2507v1