[Paper Review] On isomorphisms of algebras of smooth functions

J. Mrčun|ArXiv.org|Sep 10, 2003

Advanced Operator Algebra Research4 references5 citations

TL;DR

This paper establishes that any isomorphism between algebras of smooth real or complex-valued functions on smooth Hausdorff manifolds (without requiring second-countability, paracompactness, or connectedness) arises uniquely from composition with a diffeomorphism between the underlying manifolds. The key innovation lies in an algebraic characterization of manifold points via 'characteristic sequences' of functions, enabling a purely algebraic proof without topological assumptions on the manifolds.

ABSTRACT

We show that for any smooth Hausdorff manifolds M and N, which are not necessarily second countable, paracompact or connected, any isomorphism from the algebra of smooth (real or complex) functions on N to the algebra of smooth functions on M is given by composition with a unique diffeomorphism from M to N. An analogous result holds true for isomorphisms of algebras of smooth functions with compact support.

Motivation & Objective

• To resolve a question posed by Alan Weinstein regarding the reconstruction of manifolds from their smooth function algebras.
• To establish a diffeomorphism reconstruction result for smooth function algebras without assuming second-countability, paracompactness, or connectedness.
• To provide a purely algebraic proof using characteristic sequences of functions, avoiding topological or spectral methods.
• To extend the result to smooth functions with compact support and to C^r and continuous function algebras.

Proposed method

• Define a 'characteristic sequence' of smooth functions at a point as a sequence satisfying f_n f_{n+1} = f_{n+1} and whose supports form a fundamental system of neighborhoods.
• Prove that such sequences exist at every point and are characterized algebraically by the intersection of supports being a singleton and at least one function having compact support.
• Show that an algebra isomorphism preserves the structure of characteristic sequences, mapping them to characteristic sequences on the other manifold.
• Use the image of a characteristic sequence under the isomorphism to define a map τ: M → N by associating the unique point whose support intersection matches.
• Prove that the isomorphism T is given by composition with τ, i.e., T(f) = f ∘ τ, using pointwise evaluation and the non-vanishing property of functions on neighborhoods.
• Establish that τ is a diffeomorphism by showing both τ and its inverse are smooth, using the preservation of smoothness under composition.

Experimental results

Research questions

• RQ1 Can the manifold structure be reconstructed purely from the algebra of smooth functions, even when the manifold lacks second-countability or paracompactness?
• RQ2 Is every algebra isomorphism between smooth function algebras on Hausdorff manifolds induced by a diffeomorphism between the underlying manifolds?
• RQ3 Can the correspondence between points and multiplicative functionals be established without topological assumptions on the manifold?
• RQ4 Does the same result hold for smooth functions with compact support, and can the proof be adapted to C^r or continuous functions?
• RQ5 Can a purely algebraic characterization of points via sequences of functions replace spectral or topological methods in this reconstruction problem?

Key findings

• Any algebra isomorphism between smooth function algebras on smooth Hausdorff manifolds arises uniquely from composition with a diffeomorphism between the manifolds.
• The diffeomorphism τ: M → N is constructed via the image of characteristic sequences under the isomorphism, ensuring well-definedness and injectivity.
• The isomorphism T is explicitly given by T(f) = f ∘ τ for all f ∈ C^∞(N, ℝ) or C^∞(N, ℂ), proving the algebraic structure determines the manifold map.
• The map τ is a diffeomorphism because both τ and its inverse are smooth, as smoothness is preserved under composition with T and T⁻¹.
• The result extends to C^r functions and continuous functions on topological manifolds, with identical proof structure.
• The proof relies solely on algebraic properties of characteristic sequences, avoiding spectral theory or topological assumptions on the manifolds.