Noncommutative vector valued $L_p$-spaces and completely $p$-summing maps

TL;DR

This paper introduces noncommutative vector-valued Lp-spaces within operator spaces and defines a new class of completely p-summing maps, extending classical notions to the operator space setting.

Contribution

It develops a framework for E-valued noncommutative Lp-spaces and introduces completely p-summing maps as their operator space analogue, generalizing previous work for p=1.

Findings

Established duality and interpolation properties for the new Lp-spaces.

Defined and characterized completely p-summing maps in the operator space context.

Extended classical p-summing map theory to noncommutative operator spaces.

Abstract

Let $E$ be an operator space in the sense of the theory recently developed by Blecher-Paulsen and Effros-Ruan. We introduce a notion of $E$-valued non commutative $L_p$-space for $1 \leq p < \infty$ and we prove that the resulting operator space satisfies the natural properties to be expected with respect to e.g. duality and interpolation. This notion leads to the definition of a ``completely p-summing" map which is the operator space analogue of the $p$-absolutely summing maps in the sense of Pietsch-Kwapie\'n. These notions extend the particular case $p=1$ which was previously studied by Effros-Ruan.