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A noncommutative maximal inequality for Fejér means on totally disconnected non-abelian groups

Fugui Ding, Guixiang Hong, Xumin Wang

Abstract

In this paper, we explore Fourier analysis for noncommutative $L_p$ space-valued functions on $G$, where $G$ is a totally disconnected non-abelian compact group. By additionally assuming that the value of these functions remains invariant within each conjugacy class, we establish a noncommutative maximal inequality for Fejér means utilizing the associated character system of $G$. This is an operator-valued version of the classical result due to Gát. We follow essentially the classical sketch, but due to the noncommutativity, many classical arguments have to be revised. Notably, compared to the classical results. the bounds of our estimates are explicity calculated.

A noncommutative maximal inequality for Fejér means on totally disconnected non-abelian groups

Abstract

In this paper, we explore Fourier analysis for noncommutative space-valued functions on , where is a totally disconnected non-abelian compact group. By additionally assuming that the value of these functions remains invariant within each conjugacy class, we establish a noncommutative maximal inequality for Fejér means utilizing the associated character system of . This is an operator-valued version of the classical result due to Gát. We follow essentially the classical sketch, but due to the noncommutativity, many classical arguments have to be revised. Notably, compared to the classical results. the bounds of our estimates are explicity calculated.
Paper Structure (8 sections, 13 theorems, 130 equations)

This paper contains 8 sections, 13 theorems, 130 equations.

Table of Contents

  1. Background and the main result
  2. Preliminaries and the key intermediate estimates
  3. Vector-valued non-commutative $L_p$-spaces
  4. Martingale structure and the key estimates
  5. Proof of Theorem \ref{['1.1']}: based on Proposition \ref{['1.3']}
  6. Proof of Proposition \ref{['1.3']}: based on Proposition \ref{['1.2']}
  7. Proof of Lemma \ref{['4.4']}
  8. Proof of Proposition \ref{['1.2']}

Key Result

Theorem 1.1

Let $f \in L_2(\mathcal{N})$. There is a universal constant $C$ such that and $\sigma_nf\rightarrow f$ bilaterally almost uniformly as $n\rightarrow\infty$.

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Lemma 2.5
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 15 more