Characterizing convex trace ranges in finite atomic von Neumann algebras

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 $K.$