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Dixmier-type traces on semifinite symmetric spaces

Galina Levitina, Alexandr Usachev

TL;DR

The paper addresses the construction and characterization of Dixmier-type traces on semifinite symmetric spaces defined via tail majorisation, $\mathcal{I}_h(\mathcal{M},\tau)$. It develops a method using $h$-compatible dilation-invariant states $\omega$ to define functionals $\tau_\omega(A)=\omega\left(t \mapsto \frac{1}{h(t)}\int_t^{\tau(\mathbf{1})} \mu(s,A)\,ds\right)$, proving these extend to normalised linear functionals on $\mathcal{I}_h(\mathcal{M},\tau)$ that respect tails. In the atomless or equal-trace atomic setting, these functionals exhaust all tail-respecting symmetric functionals, establishing a semifinite analogue of the Dixmier-trace framework. The paper also gives precise asymptotic criteria for the existence of non-trivial tail-respecting functionals in terms of the function $h$, namely conditions on $\lim_{t\to\infty} h(t)$ or $\lim_{t\to0} h(t)$ and the limsup of $h(2t)/h(t)$. Together, these results generalize the noncommutative integral paradigm to a broad class of semifinite symmetric spaces and clarify when such traces exist.

Abstract

We prove that a normalised linear functional on certain semifinite symmetric spaces respects tail majorisation if and only if it is a Dixmier-type trace.

Dixmier-type traces on semifinite symmetric spaces

TL;DR

The paper addresses the construction and characterization of Dixmier-type traces on semifinite symmetric spaces defined via tail majorisation, . It develops a method using -compatible dilation-invariant states to define functionals , proving these extend to normalised linear functionals on that respect tails. In the atomless or equal-trace atomic setting, these functionals exhaust all tail-respecting symmetric functionals, establishing a semifinite analogue of the Dixmier-trace framework. The paper also gives precise asymptotic criteria for the existence of non-trivial tail-respecting functionals in terms of the function , namely conditions on or and the limsup of . Together, these results generalize the noncommutative integral paradigm to a broad class of semifinite symmetric spaces and clarify when such traces exist.

Abstract

We prove that a normalised linear functional on certain semifinite symmetric spaces respects tail majorisation if and only if it is a Dixmier-type trace.
Paper Structure (3 sections, 3 theorems, 49 equations)

This paper contains 3 sections, 3 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main result

Key Result

Theorem 3.2

Let $h\in \Omega$ and let $\mathcal{I}_h(\mathcal{M},\tau)$ be as in Definition def_I_h. For every $h$-compatible dilation invariant state $\omega$ on $L_\infty(0,\tau(\mathbf{1}))$ the functional extends to a normalised linear functional on $\mathcal{I}_h(\mathcal{M}, \tau)$ that respects tails.

Theorems & Definitions (14)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • ...and 4 more