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Positive isometric Fourier multipliers on non-commutative $L^p$-spaces

Christoph Kriegler, Christian Le Merdy, Safoura Zadeh

Abstract

For a locally compact group \(G\), let \(\mathcal{L}G\) denote its left group von Neumann algebra and let \(L^p(\mathcal{L}G)\), \(1 \le p \le \infty\), be the corresponding non-commutative \(L^p\)-space. Given \(φ\in L^\infty(G)\), we study the Fourier multiplier \(M_{φ,p}\) acting on \(L^p(\mathcal{L}G)\). We prove that for any \(p \neq 2\), the operator \(M_{φ,p}\) is a positive surjective isometry if and only if \(φ\) coincides locally almost everywhere with a continuous character of \(G\). This characterization extends results obtained recently (jointly with C.~Arhancet) in the unimodular setting.

Positive isometric Fourier multipliers on non-commutative $L^p$-spaces

Abstract

For a locally compact group , let denote its left group von Neumann algebra and let \(L^p(\mathcal{L}G)\), , be the corresponding non-commutative -space. Given \(φ\in L^\infty(G)\), we study the Fourier multiplier acting on \(L^p(\mathcal{L}G)\). We prove that for any , the operator is a positive surjective isometry if and only if coincides locally almost everywhere with a continuous character of . This characterization extends results obtained recently (jointly with C.~Arhancet) in the unimodular setting.
Paper Structure (9 sections, 18 theorems, 153 equations)

This paper contains 9 sections, 18 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Group von Neumann algebras and their non-commutative $L^p$-spaces
  3. The weighted von Neumann algebras $({\mathcal{L}}G,\varphi_0)$ and $({\mathcal{R}}G,\psi_0)$
  4. The spaces $L^p({\mathcal{L}}G)$
  5. Some dense subspaces
  6. Bounded Fourier multipliers on $L^p({\mathcal{L}}G)$
  7. Extension properties
  8. A characterization of positive isometric Fourier multipliers
  9. The cases $p=1$ and $p=2$

Key Result

Lemma 2.1

Let $1 \leq p < \infty$.

Theorems & Definitions (36)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Definition 3.1
  • Corollary 3.3
  • proof
  • Proposition 3.5
  • proof
  • Theorem 4.1
  • proof
  • ...and 26 more