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Integration of virtually continuous functions over bistochastic measures and the trace formula for nuclear operators


Authors: A. M. Vershik, P. B. Zatitskiĭ and F. V. Petrov
Translated by: F. Petrov
Original publication: Algebra i Analiz, tom 27 (2015), nomer 3.
Journal: St. Petersburg Math. J. 27 (2016), 393-398
MSC (2010): Primary 28A25
DOI: https://doi.org/10.1090/spmj/1394
Published electronically: March 30, 2016
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Abstract: Birman's definition of the integral trace of a nuclear operator as an integral over the diagonal is linked to the recent concept of virtually continuous measurable functions of several variables [2, 3]. Namely, it is shown that the construction of Birman is a special case of the general integration of virtually continuous functions over polymorphisms (or bistochastic measures), which in particular makes it possible to integrate such functions over some submanifolds of zero measure. Virtually continuous functions have similar application to embedding theorems (see [2]).


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Additional Information

A. M. Vershik
Affiliation: St. Petersburg Branch, Steklov Institute of Mathematics, Russian Academy of Sciences; St. Petersburg State University; A. A. Kharkevich Institute for Information Transition Problems, Russian Academy of Sciences
Email: vershik@pdmi.ras.ru

P. B. Zatitskiĭ
Affiliation: St. Petersburg Branch, Steklov Institute of Mathematics, Russian Academy of Sciences; Chebyshev Laboratory at St. Petersburg State University
Email: paxa239@yandex.ru

F. V. Petrov
Affiliation: St. Petersburg Branch, Steklov Institute of Mathematics, Russian Academy of Sciences; St. Petersburg State University
Email: fedyapetrov@gmail.com

DOI: https://doi.org/10.1090/spmj/1394
Keywords: Virtally continuous function, quasibistochastic measures, duality
Received by editor(s): September 5, 2014
Published electronically: March 30, 2016
Additional Notes: Supported by the RFBR grants 14-01-00373_A and 13-01-12422-OFI-m; President of Russia grant MK-6133.2013.1; Government of Russia grant 11.G34.31.0026 (Chebyshev Laboratory); JSC “GazpromNeft”; SPbSU grant, project 6.38.223.2014.
Dedicated: To Nina Nikolaevna Ural’tseva on the occasion of her anniversary
Article copyright: © Copyright 2016 American Mathematical Society