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Generators of quantum stochastic flows and cyclic cohomology

Published online by Cambridge University Press:  01 March 1998

PAULA BEAZLEY COHEN  and
R. L. HUDSON
PAULA BEAZLEY COHEN
Affiliation:
School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA, and URA Géométrie - Analyse - Topologie, Université des Sciences et Technologies de Lille, 59655 Villeneuve d'Ascq cedex, France; e-mail: pcohengat.univ-lille1.fr
R. L. HUDSON
Affiliation:
Mathematics Department, University of Nottingham, Nottingham, NG7 2RD and Mathematical Institute, Slovak Academy of Sciences, Stefanikova 49, 81473 Bratislava, Slovakia; e-mail: rlhmaths.nott.ac.uk
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Abstract

This paper relates cyclic cohomology, as obtained from traces with suitable domains on free product algebras, to the construction, by using the quantum stochastic calculus of Hudson and Parthasarathy, of quantum (stochastic) flows. As an application of the methods of the present article, we show how the characters of finitely summable Fredholm modules, as constructed by Connes, can be recovered from quantum flows in both the specialisation to the pure gauge and to the Brownian case. We also relate the regularized trace of Connes, used in the construction of these characters, to Lindblad generators.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 123 , Issue 2 , March 1998 , pp. 345 - 363
Copyright
Cambridge Philosophical Society 1998

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Footnotes

This work was in part supported by EU HCM, contract CHRX-CT93-OD94. The first author (PBC) was also supported by the Institute for Advanced Study, Princeton, New Jersey, USA, NSF grant number DMS 9304580, by the Centre National de Recherche Scientifique, Lille, France and by the Max-Planck-Institut für Mathematik, Bonn, Germany.