Ranges of bimodule projections and reflexivity

TL;DR

This paper investigates conditions under which the weak* closed linear span of two reflexive masa-bimodules remains reflexive, introducing new classes of bimodules and linking operator synthesis with strong operator Ditkin.

Contribution

It develops a framework for reflexivity in dual Banach spaces and characterizes operator synthetic classes of masa-bimodules via new projections.

Findings

Affirmative conditions for reflexivity of bimodule spans

Introduction of new classes of masa-bimodules based on projections

Equivalence of operator synthesis and strong operator Ditkin for studied bimodules

Abstract

We develop a general framework for reflexivity in dual Banach spaces, motivated by the question of when the weak* closed linear span of two reflexive masa-bimodules is automatically reflexive. We establish an affirmative answer to this question in a number of cases by examining two new classes of masa-bimodules, defined in terms of ranges of masa-bimodule projections. We give a number of corollaries of our results concerning operator and spectral synthesis, and show that the classes of masa-bimodules we study are operator synthetic if and only if they are strong operator Ditkin.