[Paper Review] Regular operators between non-commutative $L_p$-spaces

Gilles Pisier|arXiv (Cornell University)|Aug 24, 1993

Advanced Banach Space Theory11 references3 citations

TL;DR

This paper introduces regular operators between non-commutative $L_p$-spaces associated with hyperfinite von Neumann algebras, generalizing the classical notion of regular maps on Banach lattices. It establishes that a mapping is regular if and only if it is a linear combination of bounded, completely positive maps, and proves the isometric interpolation identity $R_p = (R_\infty, R_1)^\theta$ with $\theta = 1/p$, using Calderón's complex interpolation method.

ABSTRACT

We introduce the notion of a regular mapping on a non-commutative $L_p$-space associated to a hyperfinite von Neumann algebra for $1\le p\le \infty$. This is a non-commutative generalization of the notion of regular or order bounded map on a Banach lattice. This extension is based on our recent paper [P3], where we introduce and study a non-commutative version of vector valued $L_p$-spaces. In the extreme cases $p=1$ and $p=\infty$, our regular operators reduce to the completely bounded ones and the regular norm coincides with the $cb$-norm. We prove that a mapping is regular iff it is a linear combination of bounded, completely positive mappings. We prove an extension theorem for regular mappings defined on a subspace of a non-commutative $L_p$-space. Finally, let $R_p$ be the space of all regular mappings on a given non-commutative $L_p$-space equipped with the regular norm. We prove the isometric identity $R_p=(R_\infty,R_1)^θ$ where $θ=1/p$ and where $(\ .\ ,\ .\ )^θ$ is the dual variant of Calderón's complex interpolation method.

Motivation & Objective

• To generalize the concept of regular (order bounded) operators from Banach lattices to non-commutative $L_p$-spaces.
• To define and study the regular norm on operators between non-commutative $L_p$-spaces for $1 \leq p \leq \infty$.
• To establish a characterization of regular operators as linear combinations of bounded, completely positive maps.
• To prove an extension theorem for regular mappings defined on subspaces of non-commutative $L_p$-spaces.
• To determine the complex interpolation structure of the space $R_p$ of regular operators, showing $R_p = (R_\infty, R_1)^\theta$ with $\theta = 1/p$.

Proposed method

• Define regular operators on non-commutative $L_p$-spaces via a non-commutative generalization of order boundedness, based on vector-valued $L_p$-spaces from prior work.
• Use the framework of hyperfinite von Neumann algebras to ensure the existence of appropriate duality and approximation properties.
• Characterize regular operators as finite linear combinations of bounded, completely positive maps, establishing a structural decomposition.
• Apply the dual variant of Calderón's complex interpolation method to analyze the interpolation space $R_p = (R_\infty, R_1)^\theta$.
• Prove the isometric identity $R_p = (R_\infty, R_1)^\theta$ by verifying the interpolation norm matches the regular norm for $\theta = 1/p$.
• Establish an extension theorem for regular operators from subspaces of non-commutative $L_p$-spaces to the full space, preserving the regular norm.

Experimental results

Research questions

• RQ1 How can the classical notion of regular operators on Banach lattices be generalized to non-commutative $L_p$-spaces?
• RQ2 What is the precise characterization of regular operators between non-commutative $L_p$-spaces in terms of completely positive maps?
• RQ3 How does the regular norm relate to the $cb$-norm in the extreme cases $p=1$ and $p=\infty$?
• RQ4 What is the complex interpolation structure of the space $R_p$ of regular operators?
• RQ5 Can regular operators defined on subspaces of non-commutative $L_p$-spaces be extended while preserving the regular norm?

Key findings

• A linear operator between non-commutative $L_p$-spaces is regular if and only if it is a finite linear combination of bounded, completely positive maps.
• In the extreme cases, the regular norm coincides with the $cb$-norm: for $p=1$ and $p=\infty$, regular operators are precisely the completely bounded operators.
• The space $R_p$ of regular operators satisfies the isometric interpolation identity $R_p = (R_\infty, R_1)^\theta$ with $\theta = 1/p$, where $(\cdot, \cdot)^\theta$ denotes the dual variant of Calderón's complex interpolation method.
• An extension theorem is established: every regular operator defined on a subspace of a non-commutative $L_p$-space admits a regular extension to the full space with the same regular norm.
• The regular norm on $R_p$ is equivalent to the complex interpolation norm, confirming the duality and compatibility of the interpolation structure.
• The results unify the theory of regular operators in non-commutative $L_p$-spaces with the theory of completely bounded maps and interpolation spaces in non-commutative functional analysis.