When are Fredholm triples operator homotopic?

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 wi...

Full description

Saved in:
Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2007-02, Vol.135 (2), p.405-415
Main Author: Kucerovsky, Dan
Format: Article
Language:English
Subjects:
Algebra
Category theory, homological algebra
Equivalence relation
Exact sciences and technology
Factorization
Homomorphisms
K-theory
Mathematical analysis
Mathematical theorems
Mathematics
Operator theory
Preprints
Sciences and techniques of general use
Subalgebras
Topological theorems
Topology
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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 ({\em 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.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-06-08481-4