C*Algebras over Topological Spaces: Filtrated KTheory
Abstract
We define the filtrated Ktheory of a C*algebra over a finite topological space X and explain how to construct a spectral sequence that computes the bivariant Kasparov theory over X in terms of filtrated Ktheory. For finite spaces with totally ordered lattice of open subsets, this spectral sequence becomes an exact sequence as in the Universal Coefficient Theorem, with the same consequences for classification. We also exhibit an example where filtrated Ktheory is not yet a complete invariant. We describe a space with four points and two C*algebras over this space in the bootstrap class that have isomorphic filtrated Ktheory but are not KK(X)equivalent. For this particular space, we enrich filtrated Ktheory by another Ktheory functor, so that there is again a Universal Coefficient Theorem. Thus the enriched filtrated Ktheory is a complete invariant for purely infinite, stable C*algebras with this particular spectrum and belonging to the appropriate bootstrap class.
 Publication:

arXiv eprints
 Pub Date:
 October 2008
 DOI:
 10.48550/arXiv.0810.0096
 arXiv:
 arXiv:0810.0096
 Bibcode:
 2008arXiv0810.0096M
 Keywords:

 Mathematics  Operator Algebras;
 Mathematics  KTheory and Homology;
 19K35;
 46L35;
 46L80;
 46M18;
 46M20
 EPrint:
 Changes to theorem and equation numbering!