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The Schur-Horn theorem for operators with finite spectrum


Authors: B. V. Rajarama Bhat and Mohan Ravichandran
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
Published electronically: July 2, 2014
MathSciNet review: 3238420
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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.


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Additional Information

B. V. Rajarama Bhat
Affiliation: Indian Statistical Institute, R V College Post, Bangalore, India 560059
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

DOI: https://doi.org/10.1090/S0002-9939-2014-12114-9
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
Article copyright: © Copyright 2014 American Mathematical Society