Bilinear Ideals in Operator Spaces

TL;DR

This paper introduces and studies various bilinear ideals of jointly completely bounded mappings in operator spaces, establishing their properties and relationships, including the identification of the integral ideal with linear completely integral mappings.

Contribution

It defines new bilinear ideals in operator spaces and explores their fundamental properties and connections to existing concepts like nuclear and integral mappings.

Findings

The ideal of completely integral mappings is identified with linear completely integral mappings.

Basic properties of bilinear ideals such as nuclear, integral, and extendible are established.

Relationships between different bilinear ideals are clarified.

Abstract

We introduce a concept of bilinear ideal of jointly completely bounded mappings between operator spaces. In particular, we study the bilinear ideals $\mathcal{N}$ of completely nuclear, $\mathcal{I }$ of completely integral, $\mathcal{E}$ of completely extendible bilinear mappings, $\mathcal{MB}$ multiplicatively bounded and its symmetrization $\mathcal{SMB}$. We prove some basic properties of them, one of which is the fact that $\mathcal{I}$ is naturally identified with the ideal of (linear) completely integral mappings on the injective operator space tensor product.