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Non-linear traces of Choquet type on AF algebras

Ryota Ninomiya

TL;DR

The paper develops a non-linear Choquet-type integration theory for unital AF algebras by linking Choquet traces to the dimension scale $\\Gamma(A)$. It proves a bijection between Choquet traces on a unital AF algebra $A$ and increasing functions $\\alpha: \\Gamma(A) \to [0,\\infty)$ with $\\alpha(0)=0$, and provides explicit Choquet formulas on each finite-dimensional block that extend compatibly along the AF filtration. The main technical achievement is a constructive, inductive-limit approach that yields a global trace $\\Phi_\\alpha$ characterized by $\\Phi_\\alpha(p) = \\alpha([p])$ for projections and satisfies unitary invariance, monotonicity, positive homogeneity, and comonotonic additivity on spectrum. The work includes concrete examples (e.g., UHF and Fibonacci AF algebras) and a method to generate non-linear traces from a given tracial state, highlighting the role of the dimension scale in shaping non-commutative Choquet-type integrals and their potential connections to $K_0$ data and operator-algebraic probability.

Abstract

We study non-linear traces of Choquet type on AF algebras. Building on the characterization of Choquet traces on matrix algebras due to Nagisa--Watatani, we generalize the construction to arbitrary unital AF algebras. We show that there is a one-to-one correspondence between such traces and increasing functions on the dimension scale, and we obtain explicit Choquet formulas in terms of the spectrum and ranks of spectral projections along a fixed AF filtration.

Non-linear traces of Choquet type on AF algebras

TL;DR

The paper develops a non-linear Choquet-type integration theory for unital AF algebras by linking Choquet traces to the dimension scale . It proves a bijection between Choquet traces on a unital AF algebra and increasing functions with , and provides explicit Choquet formulas on each finite-dimensional block that extend compatibly along the AF filtration. The main technical achievement is a constructive, inductive-limit approach that yields a global trace characterized by for projections and satisfies unitary invariance, monotonicity, positive homogeneity, and comonotonic additivity on spectrum. The work includes concrete examples (e.g., UHF and Fibonacci AF algebras) and a method to generate non-linear traces from a given tracial state, highlighting the role of the dimension scale in shaping non-commutative Choquet-type integrals and their potential connections to data and operator-algebraic probability.

Abstract

We study non-linear traces of Choquet type on AF algebras. Building on the characterization of Choquet traces on matrix algebras due to Nagisa--Watatani, we generalize the construction to arbitrary unital AF algebras. We show that there is a one-to-one correspondence between such traces and increasing functions on the dimension scale, and we obtain explicit Choquet formulas in terms of the spectrum and ranks of spectral projections along a fixed AF filtration.
Paper Structure (5 sections, 9 theorems, 137 equations)

This paper contains 5 sections, 9 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Choquet integrals and Choquet traces
  3. Non-linear traces on finite direct sums of matrix algebras of Choquet type
  4. Non-linear traces on AF algebras of Choquet type
  5. Examples

Key Result

Lemma 2.2

Let $A$ be a unital C$^*$-algebra and $\varphi:A^+\to[0,\infty)$ be a Choquet trace with $\varphi(0)=0$. Then for all $a,b\in A^+$, In particular, $\varphi$ is norm continuous on $A^+$.

Theorems & Definitions (23)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • ...and 13 more