Uniform Closure of Dual Banach Algebras

We give a characterization of the ”uniform closure” of the dual of a C-algebra. Some applications in harmonic analysis are given. 1. Four topologies on the unit ball of a C-algebra Let A be a C-algebra and P (A),S(A),A1, and A denote the pure state space, state space, closed dual unit ball, and the dual of A, respectively, all equipt with the w-topology. Let A1denote the closed unit ball of A. Following [L], we consider the following four topologies on A1: 1. (w) ai → 0 iff < ai, f >→ 0 (f ∈ A), 2. (wo) ai → 0 iff < aix, f >→ 0 (x ∈ A, f ∈ A), 3. (so) ai → 0 iff ||aix|| → 0 (x ∈ A), 4. (uc) ai → 0 iff ai → 0 , uniformely on w-compact subsets of S(A). Note that (so) is just the restriction of the strict topology of the multiplier algebra M(A) to A1. Also note that if in (4) one requires the uniform convergence on w-compact subsets of A1 (instead of S(A) ), one get nothing but the norm topology (Banach-Alaoglu). Lemma 1.1. (Akemann-Glimm) Let H be a Hilbert space and S, T ∈ B(H). Let g : R → R be a non negative Borel function and 0 < θ < 1. Assume moreover that 1. 0 ≤ T ≤ 1, S = S, and S ≥ T , 2. g ≥ 1 on [θ,+∞) , 3. < Tζ, ζ >≥ 1− θ, for some ζ ∈ H. Then < g(S)ζ, ζ >≥ 1− 4 √ θ. Proof. Lemma 11.4.4 of [D]. Proposition 1.1. Topologies w, wo, and so coincide on A1 and they are stronger than uc. Proof. (wo⊂w). Given f ∈ A and x ∈ A , consider the Arens product x.f ∈ A defined by x.f(a) = f(ax) (a ∈ A) . If {ai} ⊂ A1 and ai → 0 (w) then < aix, f >=< ai, x.f >→ 0 , i.e. ai → 0 (wo). (w⊂wo). By the Cohen Factorization theorem [DW] , we have A = A.A = {x.f : x ∈ A, f ∈ A∗}. Now if {ai} ⊂ A1 and ai → 0 (wo), then given g ∈ A , choose x ∈ A and f ∈ A such that g = x.f . Then < ai, g >=< aix, f >→ 0 , i.e. ai → 0 (w). (wo⊂so). Trivial. 1991 Mathematics Subject Classification. Primary 46L05: Secondary 22D25.