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

TL;DR

This paper introduces filtrated K-theory for C*-algebras over finite topological spaces, establishing spectral sequences and invariants for classification, with examples showing limitations and enhancements for completeness.

Contribution

It develops filtrated K-theory and spectral sequences for C*-algebras over finite spaces, and proposes an enrichment to achieve a complete classification invariant.

Findings

Spectral sequence computes Kasparov theory over X.

Filtrated K-theory is not always complete as an invariant.

Enrichment of filtrated K-theory yields a complete invariant for certain spaces.

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 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 a space with four points and two C*-algebras over this space in the bootstrap class that have isomorphic filtrated K-theory but are not KK(X)-equivalent. For this particular space, we enrich filtrated K-theory by another K-theory functor, so that there is again a Universal Coefficient Theorem. Thus the enriched filtrated K-theory is a complete invariant for purely infinite, stable C*-algebras with this particular spectrum and belonging to the appropriate bootstrap class.