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Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the distinction between the classes of Dixmier and Connes-Dixmier traces


Authors: Fedor Sukochev, Alexandr Usachev and Dmitriy Zanin
Journal: Proc. Amer. Math. Soc. 141 (2013), 2169-2179
MSC (2010): Primary 58B34, 46L52
Published electronically: December 28, 2012
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Abstract: We prove that the classes of Dixmier and Connes-Dixmier traces differ even on the Dixmier ideal $ \mathcal M_{1,\infty }$. We construct a Marcinkiewicz space $ \mathcal M_\psi $ and a positive operator $ T\in \mathcal M_\psi $ which is Connes-Dixmier measurable but which is not Dixmier measurable.


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Additional Information

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

Alexandr Usachev
Affiliation: School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia
Email: a.usachev@unsw.edu.au

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

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11853-2
PII: S 0002-9939(2012)11853-2
Keywords: Dixmier trace, Marcinkiewicz space, generalized limits.
Received by editor(s): September 22, 2011
Published electronically: December 28, 2012
Additional Notes: The authors’ research was supported by the Australian Research Council
Communicated by: Varghese Mathai
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.