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1. CJM 2012 (vol 64 pp. 368)
C$^*$-Algebras over Topological Spaces: Filtrated K-Theory 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 Category:19K35 |