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

Abstract

We present a natural imbedding of the crossed product $\mathcal{A}{⋊}_{\xi }{\mathbb{Z}}^d$ 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 $\mathcal{A}{⋊}_{\xi }{\mathbb{Z}}^d$ produces generalized Fredholm operators over ${\mathcal{H}}_{\mathcal{A}}$ and Mingo's index defines a $KK$-map from $K_0(\mathcal{A}{⋊}_{\xi }{\mathbb{Z}}^d)$ 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 $K_0(\mathcal{A}{⋊}_{\xi }{\mathbb{Z}}^d)$. 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.

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References in corpus (2)

• Topological insulators and superconductors: ten-fold way and dimensional hierarchy
• Index theory for locally compact noncommutative geometries

Cited by in corpus (4)

• Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics
• Controlled topological phases and bulk-edge correspondence
• The $K$-theoretic bulk-edge correspondence for topological insulators
• Generalized Connes-Chern characters in KK-theory with an application to weak invariants of topological insulators