Research
Work
Isomorphisms in K-theory from isomorphisms in groupoid homology theories, arXiv preprint.
Ample groupoid homology and étale correspondences, arXiv preprint. To appear in the Journal of Noncommutative Geometry.
Functors between Kasparov categories from étale groupoid correspondences, arXiv preprint.
K-theory for étale groupoid C*-algebras via groupoid correspondences and spectral sequences, PhD thesis, available at this webpage.
Overview
C*-algebras are algebras of operators whose study has a topological flavour, where K-theory is an important¹ algebraic invariant. Étale groupoids are a type of topological dynamical system from which we can construct C*-algebras, covering an extremely large class² of C*-algebras. I am interested in the K-theory of the C*-algebras associated to étale groupoids.
I work with groupoid correspondences to better understand this process.
Groupoid correspondences
There are difficulties with turning the construction of a C*-algebra from an étale groupoid into a functorial construction. The construction is covariant for groups but contravariant for spaces, so we cannot naïvely construct a *-homomorphism of C*-algebras from a homomorphism of groupoids. However, a correspondence of étale groupoids (introduced by Rohit Holkar) induces a correspondence of their C*-algebras. These can be viewed as "one way" or "non-invertible" version of Morita equivalences. In particular, proper correspondences induce a map in K-theory.
The ABC spectral sequence
One tool that helps us to understand the K-theory of an étale groupoid is the ABC spectral sequence, introduced by Ralf Meyer. Valerio Proietti and Makoto Yamashita studied this in the setting of ample groupoids with torsion-free stabilisers. For such groupoids satisfying a condition related to the Baum-Connes conjecture, the ABC spectral sequence says that the K-theory of the groupoid C*-algebra is built out of the groupoid homology (in a precise sense that is known as convergence of a spectral sequence).
In low dimensions, this has been applied to verify Matui's HK conjecture that for certain ample groupoids the K-theory groups are a direct sum of the homology groups of the same parity. In higher dimensions, it is more difficult to transfer homological data to K-theoretic data directly from the spectral sequence, as we don't know how the pieces are put together to build the K-theory groups. However, a morphism of spectral sequences is able to convert an isomorphism of homological data into an isomorphism of K-theory groups even in higher dimensions. My work allows us to build a morphism of ABC spectral sequences from a proper groupoid correspondence, and we obtain a functorial version of the above diagram.
Applications
Wielding a tool designed to turn proper groupoid correspondences into morphisms of spectral sequences, I am on the market for interesting examples of groupoid correspondences! Please get in touch with me if you come across any.
One interesting example of a groupoid correspondence comes from an inverse semigroup S. This acts both on its set of non-zero idempotents and on the space of filters of these idempotents, from which we obtain two groupoids. Each idempotent corresponds to a compact open subset of the space of filters, which allows us to construct a proper groupoid correspondence. The induced morphism of spectral sequences enables us to prove a K-theory formula for the inverse semigroup C*-algebra C*(S), of the kind found in this paper by Xin Li.