C*-bialgebra Defined as the Direct Sum of UHF Algebras

Abstract

Let 𝒜0(*) denote the direct sum of a certain set of uniformly hyperfinite (UHF) algebras, and let 𝒜(*) ≡ C ⊕ 𝒜0(*). We introduce a non-cocommutative comultiplication Δϕ on 𝒜(*), and give an example of comodule-C*-algebra of the C*-bialgebra (𝒜(*), Δϕ). With respect to Δϕ, we define a nonsymmetric tensor product of *-representations of UHF algebras and show tensor product formulas of Gel'fand–Na\\uımark–Segal (GNS) representations by product states.