Banach space properties of strongly tight uniform algebras
We use the work of J. Bourgain to show that some uniform algebras of analytic functions have certain Banach space properties. If X is a Banach space, we say X is strong} if X and X* have the Dunford-Pettis property, X has the Pełczyński property, and X* is weakly sequentially complete. Bourgain has shown that the ball-algebras and the polydisk-algebras are strong Banach spaces. Using Bourgain's methods, Cima and Timoney have shown that if K is a compact planar set and A is R(K) or A(K), then A and A* have the Dunford-Pettis property. Prior to the work of Bourgain, it was shown independently by Wojtaszczyk and Delbaen that R(K) and A(K) have the Pełczyński property for special classes of sets K. We show that if A is R(K) or A(K), where K is arbitrary, or if A is A(D) where D is a strictly pseudoconvex domain with smooth $C^2$ boundary in $ℂ^n$, then A is a strong Banach space. More generally, if A is a uniform algebra on a compact space K, we say A is strongly tight} if the Hankel-type operator $S_g: A → C/A$ defined by f → fg + A is compact for every g ∈ C(K). Cole and Gamelin have shown that R(K) and A(K) are strongly tight when K is arbitrary, and their ideas can be used to show A(D) is strongly tight for the domains D considered above. We show strongly tight uniform algebras are strong Banach spaces.