Characterizing convex trace ranges in finite atomic von Neumann algebras

TL;DR

This paper characterizes when the trace range in finite atomic von Neumann algebras is convex, providing necessary and sufficient conditions and exploring related sequence representations and extreme points.

Contribution

It establishes a precise criterion for convex trace ranges in finite atomic von Neumann algebras and analyzes associated sequence properties and extreme points.

Findings

Necessary and sufficient condition for convex trace ranges.

Representation of real numbers via sequence sums with binary coefficients.

Description of extreme points of the set of sequences satisfying the key property.

Abstract

The paper is devoted to characterizing convex trace ranges in finite atomic von Neumann algebras. The main result provides us with the necessary and sufficient condition for the range of a faithful normal trace on a finite atomic von Neumann algebra to be convex. In order to prove this result we will prove the following result, which has independent interest. Let ${\bf a}=(a_1, \ldots, a_n, \ldots)$ be a non-increasing positive sequence such that $\sum\limits_{n=1}^\infty a_n=1.$ Then each real number $0\le r \le 1$ can be represented in the form \( r=\sum\limits_{n=1}^\infty \varepsilon_n a_n, \,\,\, \varepsilon_n \in \{0,1\}, n\ge 1, \) if and only if the sequence ${\bf a}$ satisfies \(a_n \le 1-\sum\limits_{k=1}^n a_k \) for all $n\ge 1.$ A set $K$ of all sequences that satisfy the last property can be represented as a convex weak-compact subset of $\ell_1 = c_0^*$. We will describe the set of all extreme points of $K.$