Pentagon equation arising from state equations of a C$^{*}$-bialgebra
arXiv:0906.2507 · doi:10.1007/s11005-010-0413-5
Abstract
The direct sum ${\cal O}_{*}$ of all Cuntz algebras has a non-cocommutative comultiplication $Î_Ï$ such that there exists no antipode of any dense subbialgebra of the C$^{*}$-bialgebra $({\cal O}_{*},Î_Ï)$. From states equations of ${\cal O}_{*}$ with respect to the tensor product, we construct an operator $W$ for $({\cal O}_{*},Î_Ï)$ such that $W^{*}$ is an isometry, $W(x\otimes I)W^{*}=Î_Ï(x)$ for each $x\in {\cal O}_{*}$ and $W$ satisfies the pentagon equation.
15 pages