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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Cuntz, Joachim / Huybrechts, Daniel / Hwang, Jun-Muk


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Unbounded Fredholm modules and double operator integrals

D. Potapov1 / F. Sukochev2

1School of Informatics and Engineering, Faculty of Science and Engineering, Flinders University of SA, Bedford Park, 5042, Adelaide, SA, Australia. e-mail:

2School of Informatics and Engineering, Faculty of Science and Engineering, Flinders University of SA, Bedford Park, 5042, Adelaide, SA, Australia. e-mail:

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 2009, Issue 626, Pages 159–185, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/CRELLE.2009.006, January 2009

Publication History

Received:
2007-01-18
Revised:
2007-10-03
Published Online:
2009-01-08

Abstract

In noncommutative geometry one is interested in invariants such as the Fredholm index or spectral flow and their calculation using cyclic cocycles. A variety of formulae have been established under side conditions called summability constraints. These can be formulated in two ways, either for spectral triples or for bounded Fredholm modules. We study the relationship between these by proving various properties of the map on unbounded self adjoint operators D given by ƒ(D) = D(1 + D 2)−1/2. In particular we prove commutator estimates which are needed for the bounded case. In fact our methods work in the setting of semifinite noncommutative geometry where one has D as an unbounded self adjoint linear operator affiliated with a semi-finite von Neumann algebra ℳ. More precisely we show that for a pair D, D 0 of such operators with DD 0 a bounded self-adjoint linear operator from ℳ and , where 𝓔 is a noncommutative symmetric space associated with ℳ, then

.

This result is further used to show continuous differentiability of the mapping between an odd 𝓔-summable spectral triple and its bounded counterpart.

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[1]
A.L. Carey, V. Gayral, A. Rennie, and F.A. Sukochev
Journal of Functional Analysis, 2012, Volume 263, Number 2, Page 383
[2]
Denis Potapov and Fedor Sukochev
Acta Mathematica, 2011, Volume 207, Number 2, Page 375

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