Twisting non-commutative $L_p$ spaces

TL;DR

This paper explores the first steps in studying twisted sums of noncommutative Lp spaces as Banach modules over von Neumann algebras, combining interpolation theory and centralizer techniques to construct and analyze new noncommutative spaces.

Contribution

It introduces a novel approach to constructing and understanding extensions of noncommutative Lp spaces using interpolation and centralizers, including the development of noncommutative Kalton-Peck spaces.

Findings

Constructed noncommutative Kalton-Peck type spaces as self-extensions of Lp.

Identified properties of these noncommutative extension spaces.

Highlighted open problems for bimodule extensions in general von Neumann algebras.

Abstract

The paper makes the first steps into the study of extensions ("twisted sums") of noncommutative $L^p$-spaces regarded as Banach modules over the underlying von Neumann algebra $\mathcal M$. Our approach combines Kalton's description of extensions by centralizers (these are certain maps which are, in general, neither linear nor bounded) with a general principle, due to Rochberg and Weiss saying that whenever one finds a given Banach space $Y$ as an intermediate space in a (complex) interpolation scale, one automatically gets a self-extension $ 0\longrightarrow Y\longrightarrow X\longrightarrow Y \longrightarrow 0. $ For semifinite algebras, considering $L^p=L^p(\mathcal M,\tau)$ as an interpolation space between $\mathcal M$ and its predual $\mathcal M_*$ one arrives at a certain self-extension of $L^p$ that is a kind of noncommutative Kalton-Peck space and carries a natural bimodule structure. Some interesting properties of these spaces are presented. For general algebras, including those of type III, the interpolation mechanism produces two (rather than one) extensions of one sided modules, one of left-modules and the other of right-modules. Whether or not one can find (nontrivial) self-extensions of bimodules in all cases is left open.