Finitely Summable $K$-homology, the Index Pairing, and Cantor Minimal Systems

TL;DR

This paper investigates index pairings in crossed-product $C^*$-algebras from Cantor minimal systems, demonstrating computability via Connes' trace formulas and establishing finitely summable $K$-homology for odometers.

Contribution

It introduces methods to compute index pairings using orbit-breaking subalgebras and shows finitely summable $K$-homology for odometer systems.

Findings

Index pairings can be computed using Connes' trace formulas.

Finitely summable $K$-homology is established for odometer systems.

The approach applies to crossed-product $C^*$-algebras from Cantor minimal actions.

Abstract

We study index pairings for crossed-product $C^*$-algebras arising from minimal actions on the Cantor set. We utilize Putnam's orbit-breaking AF-subalgebras and embeddings to show we can compute any index pairing for Cantor minimal system crossed products using Connes' trace formulas. In the case of odometers, we show that the associated algebras have uniformly finitely summable $K$-homology.