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.
Categories: math.OA math.FA