The C(X)-algebra of a net and index theory

TL;DR

This paper constructs a C(X)-algebra from a precosheaf of C*-algebras over a topological space and explores its representation theory and K-homology, linking algebraic quantum field theory to geometric topology.

Contribution

It introduces a functorial method to assign C(X)-algebras to precosheaves and interprets their representation theory and K-homology in geometric terms.

Findings

Provides a geometric description of quantum field theory representations influenced by topology

Establishes a functorial construction linking precosheaves to C(X)-algebras

Connects algebraic and topological invariants via K-homology

Abstract

Given a connected and locally compact Hausdorff space X with a good base K we assign, in a functorial way, a C(X)-algebra to any precosheaf of C*-algebras A defined over K. Afterwards we consider the representation theory and the Kasparov K-homology of A, and interpret them in terms, respectively, of the representation theory and the K-homology of the associated C(X)-algebra. When A is an observable net over the spacetime X in the sense of algebraic quantum field theory, this yields a geometric description of the recently discovered representations affected by the topology of X.