Schur idempotents and hyperreflexivity

TL;DR

This paper investigates the structure of Schur idempotents with hyperreflexive ranges, demonstrating their lattice properties and how hyperreflexivity is preserved under certain algebraic operations involving tensor products and sums.

Contribution

It establishes that the set of Schur idempotents with hyperreflexive ranges forms a Boolean lattice and proves hyperreflexivity preservation under sums and tensor products involving these spaces.

Findings

The set of Schur idempotents with hyperreflexive ranges is a Boolean lattice.

Weak* closed spans of hyperreflexive spaces and ternary masa-bimodules are hyperreflexive.

Tensor products of hyperreflexive spaces and ranges of Schur idempotents are hyperreflexive.

Abstract

We show that the set of Schur idempotents with hyperreflexive range is a Boolean lattice which contains all contractions. We establish a preservation result for sums which implies that the weak* closed span of a hyperreflexive and a ternary masa-bimodule is hyperreflexive, and prove that the weak* closed span of finitely many tensor products of a hyperreflexive space and a hyperreflexive range of a Schur idempotent (respectively, a ternary masa-bimodule) is hyperreflexive.