A Serre–Swan theorem for bundles of bounded geometry

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 Operator ⁎-modules are generalizations of Hilbert C⁎-modules where the category of C⁎-algebras has been replaced by a more flexible category of involutive algebras of bounded operators: The operator ⁎-algebras. Operator ⁎-modules play an important role in the study of the unbounded Kasparov product.