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On geometric properties of $\ell^{p}$-spaces on unitary duals of compact groups

Meiram Akhymbek, Michael Ruzhansky

TL;DR

The paper develops a comprehensive geometric analysis of noncommutative $\ell^{p}$-spaces on the unitary dual $\widehat{G}$ of a compact group, focusing on spaces built from Schatten-von Neumann ideals. It establishes uniform convexity and smoothness, Clarkson-type inequalities, Kadec-Klee properties, and the type/cotype structure, supported by duality and complex interpolation results. By treating both the Schatten-based spaces $\ell_{sch}^{p}(\widehat{G})$ and the Hilbert-Schmidt-based spaces $\ell^{p}(\widehat{G})$, the work extends classical $L^{p}$-space geometry to a noncommutative setting tied to representation theory and noncommutative harmonic analysis. These results provide a robust framework for understanding the reflexivity, duality, and interpolation behavior of noncommutative $\ell^{p}$-spaces on unitary duals, with potential applications to noncommutative Hausdorff-Young-type inequalities and related operator-ideal structures.

Abstract

In this paper, we study geometric properties of $\ell^{p}$-spaces associated with the unitary dual of a compact group. More precisely, we prove uniform smoothness, uniform convexity, Clarkson type inequalities, Kadec-Klee property, as well as type and cotype properties of such spaces. We also present duality and complex interpolation results.

On geometric properties of $\ell^{p}$-spaces on unitary duals of compact groups

TL;DR

The paper develops a comprehensive geometric analysis of noncommutative -spaces on the unitary dual of a compact group, focusing on spaces built from Schatten-von Neumann ideals. It establishes uniform convexity and smoothness, Clarkson-type inequalities, Kadec-Klee properties, and the type/cotype structure, supported by duality and complex interpolation results. By treating both the Schatten-based spaces and the Hilbert-Schmidt-based spaces , the work extends classical -space geometry to a noncommutative setting tied to representation theory and noncommutative harmonic analysis. These results provide a robust framework for understanding the reflexivity, duality, and interpolation behavior of noncommutative -spaces on unitary duals, with potential applications to noncommutative Hausdorff-Young-type inequalities and related operator-ideal structures.

Abstract

In this paper, we study geometric properties of -spaces associated with the unitary dual of a compact group. More precisely, we prove uniform smoothness, uniform convexity, Clarkson type inequalities, Kadec-Klee property, as well as type and cotype properties of such spaces. We also present duality and complex interpolation results.
Paper Structure (12 sections, 21 theorems, 213 equations)

This paper contains 12 sections, 21 theorems, 213 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Strongly continuous irreducible unitary representations
  4. Noncommutative $\ell^{p}$-spaces associated with a compact group
  5. Uniformly convex and uniformly smooth Banach spaces, type and co-type properties
  6. Complex interpolation
  7. Family of $\ell^{p}$-spaces $\ell_{sch}^{p}(\widehat{G})$ based on Schatten-von Neumann ideals
  8. Duality results
  9. Complex interpolation of $\ell_{sch}^{p}(\widehat{G})$ spaces.
  10. Clarkson's inequalities and Kadec-Klee property
  11. Geometric properties of $\ell_{sch}^{p}(\widehat{G})$
  12. Family of $\ell^{p}$-spaces $\ell^{p}(\widehat{G})$ based on Hilbert-Schmidt ideal

Key Result

Proposition 2.3

fischer2016quantization For $1\leq p\leq 2$, one has a continuous embedding $\ell^{p}(\widehat{G})\hookrightarrow \ell_{sch}^{p}(\widehat{G})$ and For $2\leq p\leq \infty$, one has a continuous embedding $\ell_{sch}^{p}(\widehat{G})\hookrightarrow\ell^{p}(\widehat{G})$ and

Theorems & Definitions (44)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 34 more