Dilations and rigid factorisations on noncommutative L^p-spaces

TL;DR

This paper investigates the limitations of dilation theorems for positive contractions in noncommutative L^p-spaces, highlighting fundamental differences from classical cases and exploring operator space analogs.

Contribution

It demonstrates the absence of a noncommutative analog of Akcoglu's dilation theorem and develops non symmetric operator space valued noncommutative L^p-spaces.

Findings

No reasonable noncommutative dilation theorem exists for positive contractions.

Development of non symmetric operator space valued noncommutative L^p-spaces.

Insights into the structure of noncommutative L^p-spaces and their dilation properties.

Abstract

We study some factorisation and dilation properties of completely positive maps on noncommutative L^p-spaces. We show that Akcoglu's dilation theorem for positive contractions on classical (=commutative) L^p-spaces has no reasonable analog in the noncommutative setting. Our study relies on non symmetric analogs of Pisier's operator space valued noncommutative L^p-spaces that we investigate in the first part of the paper.