A characterization of reflexive spaces of operators

TL;DR

This paper characterizes reflexive spaces of operators using bilattice maps, extending known results about weakly closed bimodules over nest algebras to a broader class of operator spaces.

Contribution

It provides a new characterization of reflexive operator spaces via order-preserving maps on bilattices, generalizing previous results for nest algebra bimodules.

Findings

Equivalence of reflexivity and bilattice map conditions.

Extension of Erdos--Power characterization to reflexive spaces.

Provides a new framework for understanding operator space reflexivity.

Abstract

We show that for a linear space of operators ${\mathcal M}\subseteq {\mathcal B}(H_1,H_2)$ the following assertions are equivalent. (i) ${\mathcal M} $ is reflexive in the sense of Loginov--Shulman. (ii) There exists an order-preserving map $\Psi=(\psi_1,\psi_2)$ on a bilattice $Bil({\mathcal M})$ of subspaces determined by ${\mathcal M}$, with $P\leq \psi_1(P,Q)$ and $Q\leq \psi_2(P,Q)$, for any pair $(P,Q)\in Bil({\mathcal M})$, and such that an operator $T\in {\mathcal B}(H_1,H_2)$ lies in ${\mathcal M}$ if and only if $\psi_2(P,Q)T\psi_1(P,Q)=0$ for all $(P,Q)\in Bil( {\mathcal M})$. This extends to reflexive spaces the Erdos--Power type characterization of weakly closed bimodules over a nest algebra.