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

http://dx.doi.org/10.4153/CJM-2011-061-x
Canad. J. Math. 64(2012), 368-408

  Published:2012-03-05
 Printed: Apr 2012

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.