Operator synthesis. I. Synthetic sets, bilattices and tensor algebras

Abstract

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...

Beurling-Pollard type theorems

“…More generally, for any subset M ⊂ B(H 1 , H 2 ) there is the smallest pseudo-closed set, supp M, which supports all operators in M (see [ShT1]). …”

Operator synthesis II: Individual synthesis and linear operator equations

“…Let W ⊂ X ×Y be a closed set supporting T . By [ShT,Theorem 4.3], given F ∈ J(W ), T, F G = 0 for any G ∈ Γ(X, Y ) and hence F · T = 0. Applying now Lemma 3.4, we obtain Supp(T ) ⊂ W , showing that Supp(T ) is the smallest closed set supporting T .…”

“…By [ShT,Theorem 4.8] the set {(x, y) | f i (x) = g i (y), 1 ≤ i ≤ n} is synthetic. As it was noticed in Remark 5.6, T is supported in {(x, y) | f i (x) = g i (y), 1 ≤ i ≤ n} and the statement now follows from Corollary 5.4.…”

Representations of group algebras in spaces of completely bounded maps

“…These all may be regarded as Banach algebras of functions on E×E by Proposition 2.1. However, as pointed out in [20], an element u of V b (E) may not be continuous on E×E, even if E is compact. Hovever, if C is a closed subalgebra of C b (E) (say C = C 0 (E)), then for each u ∈ C ⊗ eh C ⊂ V b (E), the pointwise slices, u(·, x) and u(x, ·) for any fixed x in E, will always be elements of C. In the case where E is a compact group, V 0 (E) is discussed in [23], and in a profound way in [25].…”