A characterization for discrete quantum group

Based on the work of A.Van Daele, E.G.Effros and Z.J.Ruan on mul-tiplier Hopf algerba and discrete quantum group, this paper states that discrete quantum group (A, Δ) is exactly the set {(ω - ⊗)Δ(a)|a ε A; ω 2 A*}, where A* is the space of all reduced functionals on A. Furthermore, this paper characterizes (A, Δ) as an algebraic quantum group with a standard *-operation and a special element z ε A such that (1 ⊗ a)Δ(z) = Δ(z)(a ⊗ 1) (∀a ε A).