Intrinsic Chern-Connes Characters for Crossed Products by $\mathbb Z^d$

TL;DR

This paper develops an intrinsic approach to Chern-Connes characters for crossed product $C^*$-algebras by $Z^d$, providing an index formula and applications to weak topological invariants in condensed matter physics.

Contribution

It introduces a canonical embedding of crossed products into operator modules, defining intrinsic Fredholm modules and deriving an index formula with applications to topological invariants.

Findings

Established an intrinsic index pairing formula for $K_0$ of crossed products.

Connected the index pairing to physical weak topological invariants.

Provided a local index formula applicable in condensed matter physics.

Abstract

We present a natural imbedding of the crossed product $\mathcal A \rtimes_\xi \mathbb Z^d$ into the $C^\ast$-algebra of adjointable operators over the standard Hilbert $\mathcal A$-module $\mathcal H_{\mathcal A}$. By replacing the representations on Hilbert spaces with this canonical imbedding, we define Fredholm modules and corresponding Chern-Connes characters that are intrinsic to the $C^\ast$-dynamical system $(\mathcal A,\xi,\mathbb Z^d)$. The compression of the Dirac operator against projectors from $\mathcal A \rtimes_\xi \mathbb Z^d$ produces generalized Fredholm operators over $\mathcal H_{\mathcal A}$ and Mingo's index defines a $KK$-map from $K_0(\mathcal A \rtimes_\xi \mathbb Z^d)$ to $K(\mathcal A)$. Using a generalized Fedosov principle and a generalized Fedosov formula, we prove an index formula for the pairing of the intrinsic Chern-Connes characters and $K_0(\mathcal A \rtimes_\xi \mathbb Z^d)$. This pairing takes values in the image of $K_0(\mathcal A)$ in $\mathbb R$ under a canonical trace. A local index formula enables new applications in condensed matter physics to the so called weak topological invariants.