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

Abstract

The direct sum ${\mathcal{O}}_{\ast }$ of all Cuntz algebras has a non-cocommutative comultiplication ${\Delta }_{\phi }$ such that there exists no antipode of any dense subbialgebra of the C${}^{\ast }$-bialgebra $({\mathcal{O}}_{\ast },{\Delta }_{\phi })$. From states equations of ${\mathcal{O}}_{\ast }$ with respect to the tensor product, we construct an operator $W$ for $({\mathcal{O}}_{\ast },{\Delta }_{\phi })$ such that $W^{\ast }$ is an isometry, $W(x\otimes I)W^{\ast }={\Delta }_{\phi }(x)$ for each $x\in {\mathcal{O}}_{\ast }$ and $W$ satisfies the pentagon equation.

arXiv comment15 pages

Cited by in corpus (1)

• Idempotent set-theoretical solutions of the pentagon equation