Noncommutative Lp modules

TL;DR

This paper introduces a new framework for noncommutative L^p modules over von Neumann algebras, generalizing Hilbert modules through an L^{p/2}-valued inner product, with implications for representation theory.

Contribution

It constructs and characterizes noncommutative L^p modules using an L^{p/2}-valued inner product, extending the theory of Hilbert modules to the noncommutative setting.

Findings

Generalizes Hilbert C*-modules to L^p modules

Provides an abstract characterization based on L^{p/2}-valued inner product

Shows that L^p bimodules are nearly trivial for p not equal to 2

Abstract

We construct classes of von Neumann algebra modules by considering ``column sums" of noncommutative L^p spaces. Our abstract characterization is based on an L^{p/2}-valued inner product, thereby generalizing Hilbert C*-modules and representations on Hilbert space. While the (single) representation theory is similar to the L^2 case, the concept of L^p bimodule (p not 2) turns out to be nearly trivial.