Duality and operator algebras.pdf

a r X i v : m a t h / 0 4 0 7 2 2 0 v 2 [ m a t h .O A ] 1 6 J u l 2 0 0 4 DUALITY AND OPERATOR ALGEBRAS DAVID P. BLECHER AND BOJAN MAGAJNA Abstract. We investigate some subtle and interesting phenomena in the du- ality 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 ex- ample, 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 7→ ax on X is weak? continuous. Applications include a new characterization of the σ-weakly closed (possibly nonunital and nonselfad- joint) 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. 1. Introduction Functional analytic questions about spaces of operators often boil down to con- siderations involving dual, or weak?, topologies. In many such calculations, the key point is to prove that certain linear functions are weak? continuous. In the present paper we offer a couple of results which ensure that a linear map be automatically continuous with respect to such topologies. In the right situation, these results can be extremely useful. A multiplier of a Banach space E is a linear map T : E → E such that there exists an isometric embedding σ : E → C(?) for a compact space ?, and a function a ∈ C(?), such that σ(Tx) = aσ(x) for all x ∈ E (see [14, Section I.3] and [6, Theorem 3.7.2]). If, further, E is a dual Banach space, we are able to show the surprising fact that T is automatically w?-continuous. It is clear how to generalize the notion of multipliers, to maps on an operator space X . By definition, an operator space is a