K-theory for étale groupoid C*-algebras via groupoid correspondences and spectral sequences
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We develop techniques that allow us to convert isomorphisms in the homology of ample groupoids into isomorphisms in the K-theory of their associated C*-algebras. We apply this to prove an "orbit-stabiliser" K-theory formula for certain inverse semigroup dynamical systems, allowing us to compute the K-theory of reduced C*-algebras of inverse semigroups and left regular algebras of finitely aligned left cancellative small categories. To apply our techniques the isomorphism in homology must be induced by a correspondence of groupoids. Correspondences are a type of morphism of groupoids which can induce maps in the K-theory of the associated C*-algebras, whose isomorphisms are Morita equivalences. We extend this induced map in operator K-theory in a categorical fashion by constructing functors between the associated equivariant Kasparov categories and natural transformations between the associated K-theory functors. We also construct a map in homology from a proper correspondence of ample groupoids. This groupoid homology is related to the operator K-theory by Proietti and Yamashita's implementation of the ABC spectral sequence. We develop general functoriality for the ABC spectral sequence, which is part of a framework developed by Meyer and Nest for a categorical approach to the Baum-Connes conjecture. From a proper correspondence of étale groupoids we obtain a morphism of associated ABC spectral sequences, allowing us to construct isomorphisms in K-theory from more tractable kinds of isomorphism.
Authors
Miller, ACollections
- Theses [4248]