Characterizing convex trace ranges in finite atomic von Neumann algebras

TL;DR

The paper tackles when the trace range of a faithful normal trace on a finite atomic von Neumann algebra is convex. It reduces the problem to a discrete sequence condition: for a normalized nonincreasing sequence ${\bf a}$ with $\|{\bf a}\|_1=1$, every $r\in[0,1]$ can be expressed as $r=\sum_{n=1}^\infty \varepsilon_n a_n$ with $\varepsilon_n\in\{0,1\}$ if and only if $a_n \le 1-\sum_{k=1}^n a_k$, and defines the weak-compact convex set $K\subset \ell_1$ of such sequences. The main result then shows that $\tau(P(\mathcal M))$ is convex precisely under this sequence-inequality condition, reducing the operator-algebraic question to a combinatorial one about sequences. A complete description of the extreme boundary ext$K$ is given: extremal sequences have a block-constant form generated by integers $k_n\ge 2$, and affine maps relate $K$ to its faces $K_m$, yielding explicit mixed-base representations for numbers in $[0,1]$. These insights connect Lyapunov-type convexity phenomena with the fine combinatorial structure of trace ranges in finite atomic settings.

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 $K.$

Theorems & Definitions (12)

• Theorem 1.1
• Theorem 2.1
• proof
• Theorem 2.2
• Proposition 2.3
• proof
• Proposition 2.4
• proof
• Proposition 2.5
• proof