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# C$^*$-Algebras over Topological Spaces: Filtrated K-Theory

Published:2012-03-05
Printed: Apr 2012
• Ralf Meyer,
Mathematisches Institut and Courant Research Centre, Georg-August Universität Göttingen, 37073 Göttingen, Germany
• Ryszard Nest,
Københavns Universitets Institut for Matematiske Fag, 2100 København, Denmark
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## Abstract

We define the filtrated K-theory 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 K-theory. 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 K-theory is not yet a complete invariant. We describe two $\mathrm{C}^*$-algebras over a space $X$ with four points that have isomorphic filtrated K-theory without being $\mathrm{KK}(X)$-equivalent. For this space $X$, we enrich filtrated K-theory by another K-theory functor to a complete invariant up to $\mathrm{KK}(X)$-equivalence that satisfies a Universal Coefficient Theorem.
 Keywords: 46L35, 46L80, 46M18, 46M20
 MSC Classifications: 19K35 - Kasparov theory ($KK$-theory) [See also 58J22]