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

TL;DR

This paper introduces a new non-cocommutative comultiplication on a C*-bialgebra formed from the direct sum of UHF algebras, and explores its implications for tensor products of *-representations.

Contribution

It defines a novel non-cocommutative comultiplication on a C*-bialgebra built from UHF algebras and studies its impact on tensor product structures of representations.

Findings

Defined a non-cocommutative comultiplication on the C*-bialgebra

Constructed a comodule-C*-algebra for the new bialgebra

Derived tensor product formulas for GNS representations

Abstract

Let ${\cal A}_{0}(*)$ denote the direct sum of a certain set of UHF algebras and let ${\cal A}(*)\equiv {\bf C}\oplus {\cal A}_{0}(*)$. We introduce a non-cocommutative comultiplication $\Delta_{\phi}$ on ${\cal A}(*)$, and give an example of comodule-C$^{*}$-algebra of the C$^{*}$-bialgebra $({\cal A}(*),\Delta_{\phi})$. With respect to $\Delta_{\phi}$, we define a non-symmetric tensor product of *-representations of UHF algebras and show tensor product formulas of GNS representations by product states.