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

Cédric Arhancet

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.

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

Abstract

We define a notion of nonassociative -space associated to a -algebra (Jordan von Neumann algebra) equipped with a normal faithful state . In the particular case of -algebras underlying von Neumann algebras, we connect these spaces to a complex interpolation theorem of Ricard and Xu on noncommutative -spaces. We also make the link with the nonassociative -spaces of Iochum associated to -algebras and the investigation of contractively complemented subspaces of noncommutative -spaces. More precisely, we show that our nonassociative -spaces contain isometrically the -spaces of Iochum and that all tracial nonassociative -spaces from -factors arise as positively contractively complemented subspaces of noncommutative -spaces.
Paper Structure (21 sections, 25 theorems, 82 equations)

This paper contains 21 sections, 25 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Preliminaries
  4. Jordan algebras
  5. Centers and factors
  6. $\mathrm{JW}$-algebras
  7. $\mathrm{JW}^*$-algebras
  8. Type $\mathrm{I}_n$
  9. Traces
  10. Jordan conditional expectations
  11. Selfadjoint maps
  12. Existence of Jordan conditional expectations
  13. Interpolation
  14. Projections
  15. Complex analysis
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $\mathcal{M}$ be a $\mathrm{JW}^*$-factor with separable predual equipped with a normal finite faithful trace $\tau$. Suppose that $1 \leqslant p < \infty$. Then the Banach space $\mathrm{L}^p(\mathcal{M},\tau)$ is isometric to a positively 1-complemented subspace of a noncommutative $\mathrm{L}

Theorems & Definitions (38)

  • Theorem 1.1
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Proposition 2.8
  • Lemma 2.9
  • ...and 28 more