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C*-Algebras over Topological Spaces: Filtrated K-Theory

Published online by Cambridge University Press:  20 November 2018

Ralf Meyer  and
Ryszard Nest
Ralf Meyer
Affiliation:
Mathematisches Institut and Courant Research Centre, Georg-August Universität Göttingen, 37073 Göttingen, Germany email: rameyer@uni-math.gwdg.de
Ryszard Nest
Affiliation:
Københavns Universitets Institut for Matematiske Fag, 2100 København, Denmark email: rnest@math.ku.dk
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Abstract

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We define the filtrated $\text{K}$-theory of a ${{\text{C}}^{*}}$-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 $\text{K}$-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 $\text{K}$-theory is not yet a complete invariant. We describe two ${{\text{C}}^{*}}$-algebras over a space $X$ with four points that have isomorphic filtrated $\text{K}$-theory without being $\text{KK}\left( X \right)$-equivalent. For this space $X$, we enrich filtrated $\text{K}$-theory by another $\text{K}$-theory functor to a complete invariant up to $\text{KK}\left( X \right)$-equivalence that satisfies a Universal Coefficient Theorem.

Keywords

19K3546L3546L8046M1846M20
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 64 , Issue 2 , 16 April 2012 , pp. 368 - 408
Copyright
Copyright © Canadian Mathematical Society 2012

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