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When are Fredholm triples operator homotopic?

Author: Dan Kucerovsky
Journal: Proc. Amer. Math. Soc. 135 (2007), 405-415
MSC (2000): Primary 47L35, 19K35, 18K33
Published electronically: August 21, 2006
MathSciNet review: 2255287
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Abstract: Fredholm triples are used in the study of Kasparov's $ KK$-groups, and in Connes's noncommutative geometry. We define an absorption property for Fredholm triples, and give an if and only if condition for a Fredholm triple to be absorbing. We study the interaction of the absorption property with several of the more common equivalence relations for Fredholm triples. In general these relations are coarser than homotopy in the norm topology. We give simple conditions for an equivalence of triples to be implemented by an operator homotopy (i.e. a homotopy with respect to the norm topology). This can be expected to have applications in index theory, as we illustrate by proving two theorems of Pimsner-Popa-Voiculescu type. We show that there is some relationship with the interesting Toms-Winter characterization of $ {\mathcal D}$-absorbing algebras, recently obtained as part of Elliott's classification program.

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Additional Information

Dan Kucerovsky
Affiliation: Department of Mathematics, University of New Brunswick, P.O. Box 4400, Fredericton, New Brunswick, Canada E3B 5A3

Received by editor(s): December 10, 2004
Received by editor(s) in revised form: August 29, 2005
Published electronically: August 21, 2006
Additional Notes: The author was funded by NSERC (Canada).
Communicated by: David R. Larson
Article copyright: © Copyright 2006 American Mathematical Society