Operator synthesis and tensor products

TL;DR

This paper investigates the preservation of operator synthesis and property $S_{\sigma}$ under various operations like sums, tensor products, and Morita equivalence, providing new intersection formulas and conditions for synthesis.

Contribution

It establishes new preservation results for property $S_{\sigma}$ and operator synthesis under sums, tensor products, and Morita equivalence, and introduces intersection formulas involving masa-bimodules.

Findings

Property $S_{\sigma}$ is preserved under weak* closed sums with finite width masa-bimodules.

An intersection formula for weak* closed spans of tensor products involving finite width masa-bimodules is established.

Finitely many unions of sets of the form $ imes $, with $$ of finite width and $$ operator synthetic, are again operator synthetic under certain conditions.

Abstract

We show that Kraus' property $S_{\sigma}$ 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 masa-bimodule 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 $\kappa \times \lambda$, where $\kappa$ is a set of finite width, while $\lambda$ is operator synthetic, is, under a necessary restriction on the sets $\lambda$, again operator synthetic. We show that property $S_{\sigma}$ is preserved under spatial Morita subordinance. En route, we prove that non-atomic ternary masa-bimodules possess property $S_{\sigma}$ hereditarily.