Completely Order Bounded Maps on Non-Commutative $L_p$-Spaces

TL;DR

This paper introduces new norms on non-commutative $L_p$-spaces and characterizes decomposable maps between them, providing conditions and counterexamples for boundedness and decomposability.

Contribution

It defines norms on tensor products of non-commutative $L_p$-spaces and establishes criteria for when linear maps are decomposable, including new bounds and counterexamples.

Findings

Decomposable maps are characterized by boundedness of tensor extensions under certain conditions.

New norms on $L_p(al M) \u2297 M_n$ are introduced.

Counterexamples show limitations of boundedness for certain $p,q$ values.

Abstract

We define norms on $L_p(\mathcal{M}) \otimes M_n$ where $\mathcal{M}$ is a von Neumann algebra and $M_n$ is the complex $n \times n$ matrices. We show that a linear map $T: L_p(\mathcal{M}) \to L_q(\mathcal{N})$ is decomposable if $\mathcal{N}$ is an injective von Neumann algebra, the maps $T \otimes Id_{M_n}$ have a common upper bound with respect to our defined norms, and $p = \infty$ or $q = 1$. For $2p < q < \infty$ we give an example of a map $T$ with uniformly bounded maps $T \otimes Id_{M_n}$ which is not decomposable.