Sheaf theory and Paschke duality

TL;DR

This paper reveals that Paschke duality connects K-homology of a space with the K-theory of a sheaf of C*-algebras, simplifying key constructions in noncommutative geometry.

Contribution

It introduces a sheaf-theoretic perspective on Paschke duality, providing a natural framework for noncommutative symbols and simplifying K-homology constructions.

Findings

Paschke's dual algebra is a sheaf of C*-algebras over X.

Sheaf perspective simplifies the association of homology classes to elliptic operators.

Streamlines the construction of assembly maps in K-homology.

Abstract

Paschke duality identifies the K-homology of a space X with the K-theory of a certain dual C*-algebra. We show that Paschke's dual algebra is in a natural way the algebra of sections of a certain sheaf of C*-algebras over X, which can be thought of as a sheaf of noncommutative symbols. This conceptually simplifies a number of constructions in K-homology, such as the association of a homology class to an elliptic operator and the construction of assembly maps.