C*-algebras over topological spaces: filtrated K-theory

Ralf Meyer; Ryszard Nest

doi:10.4153/CJM-2011-061-x

C*-algebras over topological spaces: filtrated K-theory

Ralf Meyer, Ryszard Nest

Institut for Matematiske Fag

25 Citationer (Scopus)

Abstract

We define the filtrated K-theory 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 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 C*-algebras over a space X with four points that have isomorphic filtrated K-theory without being KK(X)-equivalent. For this space X, we enrich filtrated K-theory by another K-theory functor to a complete invariant up to KK(X)-equivalence that satisfies a Universal Coefficient Theorem.

Originalsprog	Engelsk
Tidsskrift	Canadian Journal of Mathematics - Journal Canadien de Mathématiques
Vol/bind	64
Udgave nummer	2
Sider (fra-til)	368-408
ISSN	0008-414X
DOI	https://doi.org/10.4153/CJM-2011-061-x
Status	Udgivet - apr. 2012

Adgang til dokumentet

10.4153/CJM-2011-061-x

Citationsformater

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AU - Nest, Ryszard

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AB - We define the filtrated K-theory 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 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 C*-algebras over a space X with four points that have isomorphic filtrated K-theory without being KK(X)-equivalent. For this space X, we enrich filtrated K-theory by another K-theory functor to a complete invariant up to KK(X)-equivalence that satisfies a Universal Coefficient Theorem.

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JO - Canadian Journal of Mathematics - Journal Canadien de Mathématiques

JF - Canadian Journal of Mathematics - Journal Canadien de Mathématiques

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