REFLEXIVITY, SUPPORTS AND SPECTRAL SYNTHESIS

The purpose of this talk is twofold. In the first part (sections 1-4) I will briefly describe the notions of generalised reflexivity and strong reflexivity for linear space of operators, as well as the problem of the density of the rank one subspace. The second part is devoted to a presentation of recent joint work with John Erdos and Victor Shulman [9] concerning reflexive subspaces admitting actions of masas. The perhaps surprising solution of the rank one density problem will be given, and a new “simultaneous co-ordinatisation ” of such subspaces will be presented. This will be given in measure-theoretic terms, and so the blanket assumption of separability of all Hilbert spaces will be made (although many results, particularly in the first part, are valid generally). The results in the first part are mostly known, apart from a few exceptions (Theorem 2.2, for example); the treatment is somewhat new1. Let me emphasise that the objects in this talk are linear spaces of operators acting on a Hilbert space. The basic notions we are interested in (reflexivity, rank one operators etc.) are not invariant under isomorphism, but only under spatial isomorphism. Thus the notion of equivalence will be unitary equivalence, not some more general notion of isomorphism. Why linear spaces and not (unital) algebras? As we will see, many results in operator algebra theory do not depend essentially on multiplication of operators; furthermore, in some problems concerning operator algebras, 1This approach is based on seminar notes which were privately circulated in 1994 under the title “Killers and preduals”. 226 linear spaces which are not algebras (such as annihilators) appear quite naturally. Thus in some instances the extra structure of multiplication (or the *-operation) may obscure the issue