Positive contractive projections on noncommutative $\mathrm{L}^p$-spaces and nonassociative $\mathrm{L}^p$-spaces

TL;DR

This paper characterizes positive contractive projections on noncommutative L^p-spaces linked to sigma-finite von Neumann algebras and connects them to nonassociative L^p-spaces associated with JW*-algebras, advancing previous results.

Contribution

It improves and refines the characterization of positive contractive projections on noncommutative L^p-spaces and relates their ranges to nonassociative L^p-spaces from JW*-algebras.

Findings

Precise characterization of positive contractive projections.

Ranges are isometric to nonassociative L^p-spaces.

Connections established with JW*-algebras.

Abstract

We continue our investigation of contractive projections on noncommutative $\mathrm{L}^p$-spaces where $1 < p < \infty$ started in \cite{ArR19}. We improve the results of \cite{ArR19} and we characterize precisely the positive contractive projections on a noncommutative $\mathrm{L}^p$-space associated with a $\sigma$-finite von Neumann algebra. We connect this topic to the theory of $\mathrm{JW}^*$-algebras. More precisely, in large cases, we are able to show that the range of a positive contractive projection is isometric to a nonassociative $\mathrm{L}^p$-space associated to a $\mathrm{JW}^*$-algebra.