C\(^*\)-algebras over topological spaces: filtrated \(K\)-theory (Q2882889)

The paper studies the so called filtrated \(K\)-theory and its applications to Kasparov's \(K\)-theory. Firstly, the authors introduce the concept of filtrated \(K\)-theory of a \(C^*\)-algebra over a finite topological space \(X\) and explain how to construct a spectral sequence that computes the \(KK\)-theory of Kasparov over \(X\) in terms of filtrated \(K\)-theory. Namely, the authors prove that the filtrated \(K\)-theory satisfies the universal coefficient theorem and is complete for \(C^*\)-algebras over those finite topological spaces with a totally ordered lattice of open subsets (Theorem 1.1). Next, the paper exhibits an example where filtrated \(K\)-theory is not yet a complete invariant (Theorem 1.2). For this concrete case, the authors enrich filtrated \(K\)-theory by another \(K\)-theory functor to a complete invariant up to \(KK\)-theoretical equivalence that satisfies a universal coefficient theorem.