Abstract
The carpenter problem in the context of II1 factors, formulated by Kadison asks: Let A ⊂ M be a masa in a type II1 factor and let E be the normal conditional expectation from M onto A. Then, is it true that for every positive contraction A in A, there is a projection P in 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.
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