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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
DOI: https://doi.org/10.1090/S0002-9939-2012-11853-2
Published electronically: December 28, 2012
MathSciNet review: 3034443
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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
MR Author ID: 229620
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
MR Author ID: 752894
Email: d.zanin@unsw.edu.au

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.