C${}^{\ast }$-bialgebra defined as the direct sum of UHF algebras

Abstract

Let ${\mathcal{A}}_0(\ast )$ denote the direct sum of a certain set of UHF algebras and let $\mathcal{A}(\ast )\equiv \mathbf{C}\oplus {\mathcal{A}}_0(\ast )$. We introduce a non-cocommutative comultiplication ${\Delta }_{\varphi }$ on $\mathcal{A}(\ast )$, and give an example of comodule-C${}^{\ast }$-algebra of the C${}^{\ast }$-bialgebra $(\mathcal{A}(\ast ),{\Delta }_{\varphi })$. With respect to ${\Delta }_{\varphi }$, we define a non-symmetric tensor product of *-representations of UHF algebras and show tensor product formulas of GNS representations by product states.

arXiv comment21 pages. arXiv admin note: substantial text overlap with arXiv:0910.1420

References in corpus (3)

• Classification and realizations of type III factor representations of Cuntz-Krieger algebras associated with quasi-free states
• A tensor product of representations of UHF algebras arising from Kronecker products
• Non-cocommutative C${}^{\ast }$-bialgebra defined as the direct sum of free group C${}^{\ast }$-algebras