We investigate some subtle and interesting phenomena in the duality theory of operator spaces and operator algebras. In particular, we give several applications of operator space theory, based on the surprising fact that certain maps are always weak * -continuous on dual operator spaces. For example, if X is a subspace of a C * -algebra A, and if a ∈ A satisfies aX ⊂ X and a * X ⊂ X, and if X is isometric to a dual Banach space, then we show that the function x → ax on X is weak * continuous. Applications include a new characterization of the σ-weakly closed (possibly nonunital and nonselfadjoint) operator algebras, and it makes possible a generalization of the theory of W * -modules to the framework of modules over such algebras. We also give a Banach module characterization of σ-weakly closed spaces of operators which are invariant under the action of a von Neumann algebra.

A Ritt operator T : X → X on Banach space is a power bounded operator satisfying an estimate n T n − T n−1 ≤ C . When X = L p (Ω) for some 1 < p < ∞, we study the validity of square functions estimates k k|T k (x)−T k−1 (x)| 2

In this paper we study the notion of joint functional calculus associated with a couple of resolvent commuting sectorial operators on a Banach space X. We present some positive results when X is, for example, a Banach lattice or a quotient of subspaces of a B‐convex Banach lattice. Furthermore, we develop a notion of a generalized H∞‐functional calculus associated with the extension to Λ(H) of a sectorial operator on a B‐convex Banach lattice Λ, where H is a Hilbert space. We apply our results to a new construction of operators with a bounded H∞‐functional calculus and to the maximal regularity problem. 1991 Mathematics Subject Classification: 47A60, 47D06, 46C15.

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