Extensions of Bundles of C*-algebras

TL;DR

This paper investigates how bundles of C*-algebras can be extended to their limits, providing existence, uniqueness, and functoriality results, with applications to the classical limit in quantum theories.

Contribution

It introduces a framework for extending bundles of C*-algebras to limiting parameters, establishing fundamental existence, uniqueness, and functorial properties.

Findings

Proves existence and uniqueness of bundle extensions.

Shows extensions are functorial for key algebraic structures.

Applies results to the classical limit in quantum mechanics.

Abstract

Bundles of C*-algebras can be used to represent limits of physical theories whose algebraic structure depends on the value of a parameter. The primary example is the $\hbar\to 0$ limit of the C*-algebras of physical quantities in quantum theories, represented in the framework of strict deformation quantization. In this paper, we understand such limiting procedures in terms of the extension of a bundle of C*-algebras to some limiting value of a parameter. We prove existence and uniqueness results for such extensions. Moreover, we show that such extensions are functorial for the C*-product, dynamical automorphisms, and the Lie bracket (in the $\hbar\to 0$ case) on the fiber C*-algebras.