Gelfand duality for manifolds, and vector and other bundles

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