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...
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).
We introduce the notion of generalized Weyl modules for twisted current algebras. We study their representation-theoretic and combinatorial properties and connection to the theory of nonsymmetric Macdonald polynomials. As an application we compute the dimension of the classical Weyl modules in the remaining unknown case.
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