The Schur-Horn theorem for operators with finite spectrum

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 in type $II_1$ factors and an approximate Schur-Horn theorem for general positive operators in type $II_1$ factors.