The Schur-Horn theorem for operators with finite spectrum

TL;DR

This paper proves the carpenter problem for positive contractions with finite spectrum in type II_1 factors and establishes exact and approximate Schur-Horn theorems for operators with finite spectrum.

Contribution

It introduces a solution to the carpenter problem for finite spectrum operators and extends Schur-Horn theorems to this setting.

Findings

Solution to the carpenter problem for finite spectrum positive contractions.

Exact Schur-Horn theorem for operators with finite spectrum.

Approximate Schur-Horn theorem for general positive operators.

Abstract

The carpenter problem in the context of $II_1$ factors, formulated by Kadison asks: Let $\mathcal{A} \subset \mathcal{M}$ be a masa in a type $II_1$ factor and let $E$ be the normal conditional expectation from $\mathcal{M}$ onto $\mathcal{A}$. Then, is it true that for every positive contraction $A$ in $\mathcal{A}$, there is a projection $P$ in $\mathcal{M}$ such that $E(P) = A$? In this note, we show that this is true if $A$ has finite spectrum. We will then use this result to prove an exact Schur-Horn theorem for (positive)operators with finite spectrum and an approximate Schur-Horn theorem for general (positive)operators.