The Schur-Horn theorem for operators with finite spectrum
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- by B. V. Rajarama Bhat and Mohan Ravichandran PDF
- Proc. Amer. Math. Soc. 142 (2014), 3441-3453 Request permission
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.References
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Additional Information
- B. V. Rajarama Bhat
- Affiliation: Indian Statistical Institute, R V College Post, Bangalore, India 560059
- MR Author ID: 314081
- Email: bhat@isibang.ac.in
- Mohan Ravichandran
- Affiliation: Department of Mathematics, Istanbul Bilgi University, Dolapdere, Istanbul, Turkey 34440
- Address at time of publication: Department of Mathematics, Mimar Sinan Fine Arts University, Bomonti, Istanbul, Turkey 34400
- Received by editor(s): February 9, 2012
- Received by editor(s) in revised form: September 4, 2012
- Published electronically: July 2, 2014
- Communicated by: Marius Junge
- © Copyright 2014 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 142 (2014), 3441-3453
- MSC (2010): Primary 46L10; Secondary 46L54
- DOI: https://doi.org/10.1090/S0002-9939-2014-12114-9
- MathSciNet review: 3238420