Operator synthesis and tensor products

Abstract

Abstract. We show that Kraus' property Sσ is preserved under taking weak* closed sums with masa-bimodules of finite width, and establish an intersection formula for weak* closed spans of tensor products, one of whose terms is a masabimodule of finite width. We initiate the study of the question of when operator synthesis is preserved under the formation of products and prove that the union of finitely many sets of the form κ × λ, where κ is a set of finite width, while λ is operator synthetic, is, under a necessary restriction on the sets λ, again operator synthetic. We show that property Sσ is preserved under spatial Morita subordinance.

Sets of multiplicity and closable multipliers on group algebras

“…With his pivotal paper [2], W. B. Arveson opened up a new avenue in that direction by introducing the notion of operator synthesis. The relation between operator synthesis and spectral synthesis for locally compact groups was explored in detail in [13], [22], [36], [9] and [10], among others. In this connection, J. Froelich [13] studied the question of when the operator algebra associated with a commutative subspace lattice contains a non-zero compact operator.…”

“…In Section 5, we examine sets of finite width. This class of sets has played a fundamental role in the field since their introduction in [2] (see [9], [10], [31] and the references therein). We characterise the sets of finite width that are also sets of operator multiplicity, and show that, in general, every compact operator supported on a set of finite width is the norm limit of sums of rank one operators supported on this set.…”

Reduced spectral synthesis and compact operator synthesis

“…Our rationale behind investigating the link between reduced spectral synthesis and compact operator synthesis is two-fold: on one hand, results from Harmonic Analysis have been highly instrumental in providing examples of operator algebras or spaces that have or fail a certain property of interest (see e.g. [2,4,12,17,18,26,46,48]); on the other hand, results obtained using operator algebraic methods have led, among others, to the identification of new classes of sets of spectral synthesis [15,16], to unification and new proofs of transference results for sets of uniqueness [54], and to the introduction of new classes of multipliers of A(G) [47].…”

Transference and preservation of uniqueness

“…In addition, operator theoretic methods have been successfully employed to obtain results belonging to the area of Harmonic Analysis per se (see e.g. [8]). These ideas, along with questions about closability of operator transformers, led to the study of sets of operator uniqueness in [19], where it was shown that a closed subset E of a second countable locally compact group G is a set of uniqueness if and only if the subset E * = {(s, t) : ts −1 ∈ E} of G × G is a set of operator uniqueness.…”