Gelfand duality for manifolds, and vector and other bundles

TL;DR

This paper extends Gelfand duality to real analytic and Stein manifolds, and further to vector, affine, and jet bundles, broadening the algebraic-topological correspondence in differential geometry.

Contribution

It generalizes Gelfand duality from smooth manifolds to more complex structures like analytic and Stein manifolds, and vector bundles, using cohomological methods.

Findings

Gelfand duality applies to real analytic and Stein manifolds.

Duality is extended to vector, affine, and jet bundles.

Unified cohomological argument underpins the generalizations.

Abstract

In general terms, Gelfand duality refers to a correspondence between a geometric, topological, or analytical category, and an algebraic category. For example, in smooth differential geometry, Gelfand duality refers to the topological embedding of a smooth manifold in the topological dual of its algebra of smooth functions. This is generalised here in two directions. First, the topological embeddings for manifolds are generalised to the cases of real analytic and Stein manifolds, using a unified cohomological argument. Second, this type of duality is extended to vector bundles, affine bundles, and jet bundles by using suitable classes of functions, the topological duals in which the embeddings take their values.