The Schur-Horn theorem for operators with finite spectrum

Abstract

The carpenter problem in the context of $I{I}_1$ factors, formulated by Kadison asks: Let $\mathcal{A}\subset \mathcal{M}$ be a masa in a type $I{I}_1$ 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.

10 pages, no figures

References in corpus (2)

• Conditional expectations onto maximal abelian *-subalgebras
• The carpenter and Schur--Horn problems for masas in finite factors