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Catalysis in the trace class and weak trace class ideals


Authors: Guillaume Aubrun, Fedor Sukochev and Dmitriy Zanin
Journal: Proc. Amer. Math. Soc. 144 (2016), 2461-2471
MSC (2010): Primary 47A80, 47B10, 47L20
Published electronically: October 20, 2015
MathSciNet review: 3477062
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Abstract: Given operators $ A,B$ in some ideal $ \mathcal {I}$ in the algebra $ \mathcal {L}(H)$ of all bounded operators on a separable Hilbert space $ H$, can we give conditions guaranteeing the existence of a trace-class operator $ C$ such that $ B \otimes C$ is submajorized (in the sense of Hardy-Littlewood) by $ A \otimes C$? In the case when $ \mathcal {I} = \mathcal {L}_1$, a necessary and almost sufficient condition is that the inequalities $ {\rm Tr} (B^p) \leq {\rm Tr} (A^p)$ hold for every $ p \in [1,\infty ]$. We show that the analogous statement fails for $ \mathcal {I} = \mathcal {L}_{1,\infty }$ by connecting it with the study of Dixmier traces.


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Additional Information

Guillaume Aubrun
Affiliation: Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France
Email: aubrun@math.univ-lyon1.fr

Fedor Sukochev
Affiliation: School of Mathematics and Statistics, University of NSW, Sydney, 2052, Australia
Email: f.sukochev@unsw.edu.au

Dmitriy Zanin
Affiliation: School of Mathematics and Statistics, University of NSW, Sydney, 2052, Australia
Email: d.zanin@unsw.edu.au

DOI: https://doi.org/10.1090/proc/12889
Received by editor(s): March 25, 2015
Received by editor(s) in revised form: June 18, 2015, and July 3, 2015
Published electronically: October 20, 2015
Additional Notes: The research of the first author was supported by the ANR projects OSQPI (ANR-11-BS01-0008) and StoQ (ANR-14-CE25-0003)
The research of the second and third authors has been supported by the ARC projects DP140100906 and DP 120103263.
Communicated by: Pamela B. Gorkin
Article copyright: © Copyright 2015 American Mathematical Society