Nonassociative $\mathrm{L}^p$-spaces and embeddings in noncommutative $\mathrm{L}^p$-spaces

TL;DR

This paper introduces a new class of nonassociative L^p-spaces linked to Jordan algebras, explores their relationship with noncommutative L^p-spaces, and demonstrates their structural properties and embeddings.

Contribution

It defines nonassociative L^p-spaces for JBW*-algebras, connects them to existing noncommutative L^p-spaces, and shows their isometric and contractive embedding properties.

Findings

Nonassociative L^p-spaces contain isometrically the L^p-spaces of Iochum.

All tracial nonassociative L^p-spaces from JW*-factors embed as contractively complemented subspaces.

The spaces are linked to complex interpolation theorems of Ricard and Xu.

Abstract

We define a notion of nonassociative $\mathrm{L}^p$-space associated to a $\mathrm{JBW}^*$-algebra (Jordan von Neumann algebra) equipped with a normal faithful state $\varphi$. In the particular case of $\mathrm{JW}^*$-algebras underlying von Neumann algebras, we connect these spaces to a complex interpolation theorem of Ricard and Xu on noncommutative $\mathrm{L}^p$-spaces. We also make the link with the nonassociative $\mathrm{L}^p$-spaces of Iochum associated to $\mathrm{JBW}$-algebras and the investigation of contractively complemented subspaces of noncommutative $\mathrm{L}^p$-spaces. More precisely, we show that our nonassociative $\mathrm{L}^p$-spaces contain isometrically the $\mathrm{L}^p$-spaces of Iochum and that all tracial nonassociative $\mathrm{L}^p$-spaces from $\mathrm{JW}^*$-factors arise as positively contractively complemented subspaces of noncommutative $\mathrm{L}^p$-spaces.