Abstract view
C$^*$Algebras over Topological Spaces: Filtrated KTheory


Published:20120305
Printed: Apr 2012
Ralf Meyer,
Mathematisches Institut and Courant Research Centre, GeorgAugust Universität Göttingen, 37073 Göttingen, Germany
Ryszard Nest,
Københavns Universitets Institut for Matematiske Fag, 2100 København, Denmark
Abstract
We define the filtrated Ktheory of a $\mathrm{C}^*$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 a 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 two $\mathrm{C}^*$algebras over a space \(X\) with four points that have isomorphic
filtrated Ktheory without being $\mathrm{KK}(X)$equivalent. For this space \(X\),
we enrich filtrated Ktheory by another Ktheory functor to a complete invariant
up to $\mathrm{KK}(X)$equivalence that satisfies a Universal Coefficient Theorem.