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

TL;DR

This paper explores the behavior of non-linear Choquet-type traces on finite factors, establishing a law of large numbers and analyzing the convergence properties of operator sequences in non-commutative probability.

Contribution

It introduces a law of large numbers for non-linear Choquet-type traces on finite factors, extending non-commutative probability theory with new convergence results.

Findings

A law of large numbers for non-linear traces was established.

Identified conditions under which averages of operators converge or have bounded accumulation points.

Discovered examples of binary shifts satisfying a uniform norm law of large numbers.

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.