A Serre-Swan theorem for bundles of bounded geometry

TL;DR

This paper generalizes the classical Serre-Swan theorem to manifolds of bounded geometry, establishing an equivalence between Hilbert bundles of bounded geometry and operator *-modules over a specific algebra.

Contribution

It extends the Serre-Swan theorem to non-compact manifolds with bounded geometry using operator *-modules and operator *-algebras.

Findings

Establishes an equivalence between Hilbert bundles of bounded geometry and operator *-modules.

Introduces the use of operator *-algebras in the context of bounded geometry.

Provides a framework for studying unbounded Kasparov products in this setting.

Abstract

The Serre-Swan theorem in differential geometry establishes an equivalence between the category of smooth vector bundles over a smooth compact manifold and the category of finitely generated projective modules over the unital ring of smooth functions. This theorem is here generalized to manifolds of bounded geometry. In this context it states that the category of Hilbert bundles of bounded geometry is equivalent to the category of operator *-modules over the operator *-algebra of continuously differentiable functions which vanish at infinity. Operator *-modules are generalizations of Hilbert C*-modules where C*-algebras have been replaced by a more flexible class of involutive algebras of bounded operators: Operator *-algebras. They play an important role in the study of the unbounded Kasparov product.