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2-positive contractive projections on noncommutative $\mathrm{L}^p$-spaces

Cédric Arhancet, Yves Raynaud

Abstract

We prove the first theorem on projections on general noncommutative $\mathrm{L}^p$-spaces associated with non-type I von Neumann algebras where $1 \leqslant p < \infty$. This is the first progress on this topic since the seminal work of Arazy and Friedman [Memoirs AMS 1992] where the problem of the description of contractively complemented subspaces of noncommutative $\mathrm{L}^p$-spaces is explicitly raised. We show that the range of a 2-positive contractive projection on an arbitrary noncommutative $\mathrm{L}^p$-space is completely order isometrically isomorphic to some noncommutative $\mathrm{L}^p$-space. This result is sharp and is even new for Schatten spaces $S^p$. Our approach relies on non-tracial Haagerup's noncommutative $\mathrm{L}^p$-spaces in an essential way, even in the case of a projection acting on a Schatten space and is unrelated to the methods of Arazy and Friedman.

2-positive contractive projections on noncommutative $\mathrm{L}^p$-spaces

Abstract

We prove the first theorem on projections on general noncommutative -spaces associated with non-type I von Neumann algebras where . This is the first progress on this topic since the seminal work of Arazy and Friedman [Memoirs AMS 1992] where the problem of the description of contractively complemented subspaces of noncommutative -spaces is explicitly raised. We show that the range of a 2-positive contractive projection on an arbitrary noncommutative -space is completely order isometrically isomorphic to some noncommutative -space. This result is sharp and is even new for Schatten spaces . Our approach relies on non-tracial Haagerup's noncommutative -spaces in an essential way, even in the case of a projection acting on a Schatten space and is unrelated to the methods of Arazy and Friedman.

Paper Structure

This paper contains 21 sections, 25 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Approach of the paper
  3. Structure of the paper
  4. Haagerup's noncommutative $\mathrm{L}^p$-spaces
  5. Duality
  6. Change of weight
  7. Case of a normal faithful linear form
  8. Reduced noncommutative $\mathrm{L}^p$-spaces
  9. Construction of suitable weight
  10. Some complements
  11. Conditional expectations on noncommutative $\mathrm{L}^p$-spaces
  12. Conditional expectations on von Neumann algebras
  13. $\mathrm{L}^p$-extensions of positive maps
  14. $\mathrm{L}^p$-extensions of conditional expectations in the general case
  15. Projections on von Neumann algebras
  16. ...and 6 more sections

Key Result

Theorem 1.1

Consider a von Neumann algebra $\mathcal{M}$. Suppose that $1 \leqslant p < \infty$. Let $P \colon \mathrm{L}^p(\mathcal{M}) \to \mathrm{L}^p(\mathcal{M})$ be a 2-positive contractive projection.

Theorems & Definitions (28)

  • Theorem 1.1: Main Theorem
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 3.1
  • Remark 3.2
  • Proposition 4.1
  • Theorem 5.1
  • Proposition 6.1
  • ...and 18 more