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Law of large numbers for non-linear traces of the Choquet type on finite factors

Masaru Nagisa, Yasuo Watatani

TL;DR

The paper extends non-linear probability into the non-commutative realm by formulating Choquet-type traces $\varphi_{\alpha}$ on II$_1$ factors and analyzing their law of large numbers. It provides a spectral-analytic representation using generalized eigenvalues and develops three extensions (symmetric, anti-symmetric, and translatable) to handle self-adjoint operators, highlighting unitary invariance, monotonicity, and triangle inequalities (under concavity). The authors establish limsup/liminf bounds for sums of operator sequences and introduce independence concepts with respect to $\varphi_{\alpha}$, applying models based on coin-tosses and Powers' binary shifts to obtain both non-commutative LLN-type results and a uniform-norm LLN in special cases. Collectively, the work advances non-linear, non-commutative probability theory on matrix algebras and type $II_1$ factors, providing a framework for understanding accumulation points of non-additive traces. $\,$

Abstract

We introduced non-linear traces of the Choquet type and the Sugeno type on semi-finite factors M in [36] as a non-commutative analog of the Choquet integral and Sugeno integral for non-additive measures. We need a weighted dimension function on the projections of M, which is an analog of a monotone measure. In this paper, we study the law of large numbers for non-linear traces of the Choquet type on finite factors M. Since averages do not converge in general, we study the range of their accumulation points, that is, we estimate their limit supremum and limit infimum. We examine the trials of sequences consisting of self-adjoint operators, which appear in coin toss or Powers' binary shifts. We have also found some unexpected examples of Powers' binary shifts which satisfy what we call the uniform norm law of large numbers. This is an attempt at non-linear and non-commutative probability theory on matrix algebras and factors of type II_1.

Law of large numbers for non-linear traces of the Choquet type on finite factors

TL;DR

The paper extends non-linear probability into the non-commutative realm by formulating Choquet-type traces on II factors and analyzing their law of large numbers. It provides a spectral-analytic representation using generalized eigenvalues and develops three extensions (symmetric, anti-symmetric, and translatable) to handle self-adjoint operators, highlighting unitary invariance, monotonicity, and triangle inequalities (under concavity). The authors establish limsup/liminf bounds for sums of operator sequences and introduce independence concepts with respect to , applying models based on coin-tosses and Powers' binary shifts to obtain both non-commutative LLN-type results and a uniform-norm LLN in special cases. Collectively, the work advances non-linear, non-commutative probability theory on matrix algebras and type factors, providing a framework for understanding accumulation points of non-additive traces.

Abstract

We introduced non-linear traces of the Choquet type and the Sugeno type on semi-finite factors M in [36] as a non-commutative analog of the Choquet integral and Sugeno integral for non-additive measures. We need a weighted dimension function on the projections of M, which is an analog of a monotone measure. In this paper, we study the law of large numbers for non-linear traces of the Choquet type on finite factors M. Since averages do not converge in general, we study the range of their accumulation points, that is, we estimate their limit supremum and limit infimum. We examine the trials of sequences consisting of self-adjoint operators, which appear in coin toss or Powers' binary shifts. We have also found some unexpected examples of Powers' binary shifts which satisfy what we call the uniform norm law of large numbers. This is an attempt at non-linear and non-commutative probability theory on matrix algebras and factors of type II_1.
Paper Structure (4 sections, 13 theorems, 177 equations)

This paper contains 4 sections, 13 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Non-linear traces of the Choquet type
  3. Three extensions of non-linear traces of the Choquet type to the self-adjoint operators
  4. Law of large numbers for non-linear traces of the Choquet type

Key Result

Proposition 2.13

(nagisawatatani4 ) Let $\mathcal{M}$ be a factor of type ${\rm II}_1$ with a faithful normal finite trace $\tau$ with $\tau(I) = 1$, and let $\alpha: [0,1] \rightarrow [0, 1]$ be a monotone increasing left continuous function with $\alpha(0) = 0$ and $\alpha(I) = 1$ . Then the non-linear trace $\va

Theorems & Definitions (58)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Example 2.8
  • Example 2.9
  • Definition 2.10
  • ...and 48 more