The interplay between the invariant subspace theory and spectral synthesis for locally compact abelian group discovered by Arveson [A] is extended to include other topics as harmonic analysis for Varopoulos algebras and approximation by projectionvalued measures. We propose a "coordinate" approach which nevertheless does not use the technique of pseudo-integral operators, as well as a coordinate free one which allows to extend to non-separable spaces some important results and constructions of [A]. We solve some problems posed in [A]. algebras, containing a masa (Arveson algebras, in terminology of [ErKSh]), by using the famous L.Schwartz's example of a non-synthetic set for the group algebra L 1 (R 3 ). Note, that among other brilliant results, [A] contains the implication M = alg L ⇒ L = lat M, for an Arveson algebra M (in full analogy with the classical situation).The results in [A] indicate, in fact, that the problematic of the operator synthesis obtains a more natural setting if instead of algebras and lattices one considers bimodules over masas and their bilattices (see the definitions below). We choose this point of view aiming at the investigation of various faces of operator synthesis, that reflect its connections with measure theory, approximation theory, linear operator equations and spectral theory of multiplication operators, synthesis in modules, Haagerup tensor products and Varopoulos tensor algebras.Let us list some results, proved in this first part of our work. We show the equivalence of several different definitions of operator synthesis. Answering a question of W.Arveson we prove the existence of a minimal pre-reflexive algebra (bimodule) with a given invariant subspace lattice (bilattice), without the assumption of separability of the underlying Hilbert space. On the other hand, for separable case we propose a coordinate approach which does not need a choice of a topology, replacing it by the pseudo-topology, naturally related to the measure spaces. This allows to consider simultaneously the synthesis for a more wide class of subsets and to avoid the use of pseudo-integral operators and the complicated theory of integral decompositions of measures (see [A] and [Da1]). This approach admits also the use of measurable sections which leads to an "inverse image theorem" (Theorem 4.7) for operator synthesis, implying in particular Arveson's theorem on synthesis for finite width lattices. We answer (in the negative) a question posed by Arveson [A][Problem, p.487] on synthesizability of the lattice generated by a synthetic lattice and a lattice of finite width (Theorem 4.9).We prove that a closed subset in a product of two compact sets is a set of spectral synthesis for the Varopoulos algebra if it is operator synthetic for any choice of measures (Theorem 6.1)(Proposition 6.1 shows that the converse implication fails). This, together with the above mentioned inverse image theorem, gives some sufficient conditions for spectral synthesis, implying, for example, the well known Drury's theorem on nontri...

Spaces of operators that are left and right modules over maximal abelian selfadjoint algebras (masa bimodules for short) are natural generalizations of algebras with commutative subspace lattices. This paper is concerned with density properties of finite rank operators and of various classes of compact operators in such modules. It is shown that every finite rank operator of a norm closed masa bimodule M is in the trace norm closure of the rank one subspace of M. An important consequence is that the rank one subspace of a strongly reflexive masa bimodule (that is, one which is the reflexive hull of its rank one operators) is dense in the module in the weak operator topology. However, in contrast to the situation for algebras, it is shown that such density need not hold in the ultraweak topology.A new method of representing masa bimodules is introduced. This uses a novel concept of an |-topology. With the appropriate notion of |-support, a correspondence is established between reflexive masa bimodules and their |-supports. It is shown that, if a C 2 -closed masa bimodule contains a trace class operator then it must contain rank one operators; indeed, every such operator is in the C 2 -norm closure of the rank one subspace of the module. Consequently the weak closure of any masa bimodule of trace class operators is strongly reflexive. However, the trace norm closure of the rank one subspace need not contain all trace class operators of the module. Also, it is shown that there exists a CSL algebra which contains no trace class operators yet contains an operator belonging to C p for all p>1. From this it follows that a transitive bimodule spanned by the rank one operators it contains need not be dense in C p for 1 p< .As an application, it is shown that there exists a commutative subspace lattice L such that L is non-synthetic but every weakly closed algebra which contains a masa and has invariant lattice L coincides with Alg L.

We investigate the connections between the invariant subspace problem for operator semigroups and the joint spectral radius. As a consequence, it is proved that any quasinilpotent Lie algebra of compact operators on a Banach space is triangularizable. We extend the Berger Wang formula to precompact sets of essentially scalar operators and prove the continuity of the joint spectral radius on them. Academic Press

Abstract. We undertake a detailed study of the sets of multiplicity in a second countable locally compact group G and their operator versions. We establish a symbolic calculus for normal completely bounded maps from the space B(L 2 (G)) of bounded linear operators on L 2 (G) into the von Neumann algebra VN(G) of G and use it to show that a closed subset E ⊆ G is a set of multiplicity if and only if the set E * = {(s, t) ∈ G × G : ts −1 ∈ E} is a set of operator multiplicity. Analogous results are established for M1-sets and M0-sets. We show that the property of being a set of multiplicity is preserved under various operations, including taking direct products, and establish an Inverse Image Theorem for such sets. We characterise the sets of finite width that are also sets of operator multiplicity, and show that every compact operator supported on a set of finite width can be approximated by sums of rank one operators supported on the same set. We show that, if G satisfies a mild approximation condition, pointwise multiplication by a given measurable function ψ : G → C defines a closable multiplier on the reduced C*-algebra C * r (G) of G if and only if Schur multiplication by the function N (ψ) : G × G → C, given by N (ψ)(s, t) = ψ(ts −1 ), is a closable operator when viewed as a densely defined linear map on the space of compact operators on L 2 (G). Similar results are obtained for multipliers on VN(G).

In this paper we study the spaces of operator-Lipschitz functions and the spaces of functions closed to them: commutator bounded. Apart from the standard operator norm on B(H), we consider a rich variety of symmetric operator norms and spaces of operator-Lipschitz functions with respect to these norms. Our approach is aimed at the investigation of the interrelation and hierarchy of these spaces and of the intrinsic properties of operator-Lipschitz functions.