Dixmier traces and extrapolation description of noncommutative Lorentz spaces

2013/02/06 by Victor Gayral, Fedor Sukochev, Gayral, Victor +1
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #math.FA #math.OA #msc:46B40 #msc:46L51 #msc:46L52 #msc:47L20

paper · pdf · doi:10.48550/arxiv.1302.1367

47 pages, final version to appear in JFA

arxiv created 2014/03/25 · arxiv updated 2014/03/26

Abstract

We study the relationships between Dixmier traces, zeta-functions and traces of heat semigroups beyond the dual of the Macaev ideal and in the general context of semifinite von Neumann algebras. We show that the correct framework for this investigation is that of operator Lorentz spaces possessing an extrapolation description. We demonstrate the applicability of our results to Hörmander-Weyl pseudo-differential calculus. In that context, we prove that the Dixmier trace of a pseudo-differential operator coincide with the `Dixmier integral' of its symbol.

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