Operator spaces which are one-sided M-Ideals in their bidual

TL;DR

This paper extends the concept of M-embedded Banach spaces to non-commutative operator spaces, showing that many classical properties are preserved in this broader setting.

Contribution

It introduces one-sided M-embedded operator spaces as non-commutative analogs and demonstrates their key properties and dual relationships.

Findings

Properties like stability under subspaces and quotients are retained.

The unique extension property is preserved.

The Radon-Nikodym Property holds in this setting.

Abstract

We generalize an important class of Banach spaces, namely the $M$-embedded Banach spaces, to the non-commutative setting of operator spaces. The one-sided $M$-embedded operator spaces are the operator spaces which are one-sided $M$-ideals in their second dual. We show that several properties from the classical setting, like the stability under taking subspaces and quotients, unique extension property, Radon Nikod$\acute {\rm{y}}$m Property and many more, are retained in the non-commutative setting. We also discuss the dual setting of one-sided $L$-embedded operator spaces.