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Self-adjoint traces on the Pedersen ideal of $\mathrm{C}^\ast$-algebras

James Gabe, Alistair Miller

TL;DR

The paper introduces $T_{\, ext{R}}(A)$, the space of self-adjoint traces on the Pedersen ideal, to address the Trace Question for non-unital $C^*$-algebras. It develops a rich duality framework, including a Kadison-type duality and a Jordan decomposition, to describe continuous traces and their extensions, with strong results for groupoid $C^*$-algebras and principled conditions ensuring the Trace Question holds. By relating $T_{\, ext{R}}(A)$ to $T_+(A)$ and affine function spaces, the authors obtain robust invariants stable under Morita equivalence and useful for classifying non-unital algebras, including a complete description of traces in principal twisted étale groupoid cases. The approach unifies traces on Pedersen ideals with $K$-theoretic pairings and invariant measures, offering practical criteria (e.g., compact base of $T_+(A)$) to guarantee affirmative answers to the Trace Question. Overall, the work broadens the landscape of trace invariants and provides concrete tools for analyzing traces in non-unital and groupoid settings.

Abstract

In order to circumvent a fundamental issue when studying densely defined traces on $\mathrm{C}^\ast$-algebras -- which we refer to as the Trace Question -- we initiate a systematic study of the set $T_{\mathbb R}(A)$ of self-adjoint traces on the Pedersen ideal of $A$. The set $T_{\mathbb R}(A)$ is a topological vector space with a vector lattice structure, which in the unital setting reflects the Choquet simplex structure of the tracial states. We establish a form of Kadison duality for $T_{\mathbb R}(A)$ and compute $T_{\mathbb R}(A)$ for principal twisted étale groupoid $\mathrm{C}^\ast$-algebras. We also answer the Trace Question positively for a large class of $\mathrm{C}^\ast$-algebras.

Self-adjoint traces on the Pedersen ideal of $\mathrm{C}^\ast$-algebras

TL;DR

The paper introduces , the space of self-adjoint traces on the Pedersen ideal, to address the Trace Question for non-unital -algebras. It develops a rich duality framework, including a Kadison-type duality and a Jordan decomposition, to describe continuous traces and their extensions, with strong results for groupoid -algebras and principled conditions ensuring the Trace Question holds. By relating to and affine function spaces, the authors obtain robust invariants stable under Morita equivalence and useful for classifying non-unital algebras, including a complete description of traces in principal twisted étale groupoid cases. The approach unifies traces on Pedersen ideals with -theoretic pairings and invariant measures, offering practical criteria (e.g., compact base of ) to guarantee affirmative answers to the Trace Question. Overall, the work broadens the landscape of trace invariants and provides concrete tools for analyzing traces in non-unital and groupoid settings.

Abstract

In order to circumvent a fundamental issue when studying densely defined traces on -algebras -- which we refer to as the Trace Question -- we initiate a systematic study of the set of self-adjoint traces on the Pedersen ideal of . The set is a topological vector space with a vector lattice structure, which in the unital setting reflects the Choquet simplex structure of the tracial states. We establish a form of Kadison duality for and compute for principal twisted étale groupoid -algebras. We also answer the Trace Question positively for a large class of -algebras.
Paper Structure (7 sections, 34 theorems, 70 equations)

This paper contains 7 sections, 34 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Continuous traces on the Pedersen ideal
  3. Jordan decomposition
  4. The dual trace space
  5. Extending traces from subalgebras
  6. Affine functions on positive traces
  7. Traces on groupoid $\mathrm{C}^\ast$-algebras

Key Result

Theorem C

Let $A$ be a $\mathrm{C}^\ast$-algebra. The canonical map $\mathrm{Ped}(A)_{\mathrm{sa}} \to T_{\mathbb R}(A)^\ast$ is a quotient map with respect to the topologies described above. Consequently, for every $f\in T_{\mathbb R}(A)^\ast$ and $\epsilon >0$ there is an $a\in \mathrm{Ped}(A)_{\mathrm{sa}}

Theorems & Definitions (77)

  • Theorem C
  • Theorem D: Duality
  • Theorem E
  • Lemma 2.1: Pedersen-book-automorphism
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Remark 2.6
  • ...and 67 more