Operator space tensor norms

TL;DR

This paper develops a systematic theory of tensor products and norms for operator spaces, highlighting differences from classical Banach space theory and exploring their interplay with mapping ideals.

Contribution

It introduces new definitions and insights for tensor norms in operator spaces, extending classical tensor product theory into the operator space setting.

Findings

New tensor norms for operator spaces are defined.

Differences between operator space and classical tensor product theories are identified.

The interplay with mapping ideals is analyzed.

Abstract

The use of a tensor product perspective has enriched functional analysis and other important areas of mathematics and physics. The context of operator spaces is clearly no exception. The aim of this manuscript is to kick off the development of a systematic theory of tensor products and tensor norms for operator spaces and its interplay with their associated mapping ideals. Based on the theory of tensor products in Banach spaces, we provide the corresponding natural definitions in the operator space framework. The theory is not a mere translation of what is known in the classical setting and new insights, techniques, ideas or hypotheses are required in many cases. As a consequence, notable differences in the theory appear when compared to the classical one.