The bounded weak star topology and the general strict topology

Abstract

Let B be a Banach algebra with an approximate identity ( e α ) such that sup ∥ e α ∥ = 1, let X be a left Banach B-module with ∥ x∥ = sup {∥ bx∥: b ϵ B, ∥ b∥ ⩽ 1}, and let β denote the strict topology induced on X by B. We show that every linear subspace of X having β-compact unit ball is a conjugate Banach space whose bounded weak star topology coincides with β. This result is applied to some common conjugate Banach spaces, namely Banach spaces with boundedly complete bases, and the spaces L p ( G) (1 < p ⩽ ∞), G a compact Abelian group. As a by-product we obtain a new representation for the strict topology on the space of bounded analytic functions on the open unit disk.