[Paper Review] Isomorphisms of algebras of smooth functions revisited

Janusz Grabowski|ArXiv.org|Oct 18, 2003

Advanced Operator Algebra Research7 references3 citations

TL;DR

This paper establishes that any algebra isomorphism between the algebras of smooth real-valued functions on Hausdorff smooth manifolds is induced by a diffeomorphism between the manifolds, without requiring second countability, paracompactness, or connectedness. The key innovation lies in characterizing points via a distinguished class of one-codimensional ideals rather than relying on multiplicative functionals, which fails for manifolds with $χ$-measurable component cardinalities.

ABSTRACT

It is proved that isomorphisms between algebras of smooth functions on Hausdorff smooth manifolds are implemented by diffeomorphisms. It is not required that manifolds are second countable nor paracompact. This solves a problem stated by A. Wienstein. Some related results are discussed as well.

Motivation & Objective

• To resolve a long-standing problem posed by A. Weinstein regarding the characterization of isomorphisms between algebras of smooth functions on smooth manifolds.
• To remove the standard assumptions of second countability, paracompactness, or connectedness in the classical result that algebra isomorphisms arise from diffeomorphisms.
• To provide a new method for identifying points in smooth manifolds via a specific class of one-codimensional ideals in the algebra of smooth functions.
• To generalize the Gel'fand-Kolmogoroff theorem on continuous function algebras to the smooth category with explicit isomorphism forms.
• To extend the result to $C^k$ and infinite-dimensional manifolds under suitable bump function conditions.

Proposed method

• Characterize points in a smooth manifold $M$ not by all multiplicative functionals but by a distinguished subclass of one-codimensional ideals in $C^∞(M;\mathbb{R})$.
• Use the existence of compactly supported smooth functions with pointwise support to define these distinguished ideals and ensure they correspond to points.
• Prove that any algebra isomorphism $\Phi: C^\infty(M_1;\mathbb{F}) \to C^\infty(M_2;\mathbb{F})$ maps such distinguished ideals to corresponding ideals, inducing a bijection between the manifolds.
• Show that this induced map $\phi: M_2 \to M_1$ is a homeomorphism by verifying that it preserves closure relations via ideal inclusions.
• Establish smoothness of $\phi$ and its inverse by using local coordinates constructed from functions in the algebra, ensuring $\phi$ is a $C^\infty$ diffeomorphism.
• Generalize the method to continuous function algebras on first-countable, completely regular spaces by constructing bump functions with point support using countable bases.

Experimental results

Research questions

• RQ1 Can the classical result that algebra isomorphisms of smooth function algebras arise from diffeomorphisms be extended to non-paracompact or non-second-countable manifolds?
• RQ2 What alternative characterization of points in a smooth manifold can replace the use of multiplicative functionals when the space lacks second countability?
• RQ3 How can the Gel'fand-Kolmogoroff theorem on continuous function algebras be adapted to the smooth category with explicit isomorphism forms?
• RQ4 What role does the cardinality of connected components play in the failure of the standard proof involving multiplicative functionals?
• RQ5 Under what conditions does the existence of smooth bump functions with point support ensure that algebra isomorphisms are pullbacks by diffeomorphisms?

Key findings

• Any algebra isomorphism between the algebras of smooth functions on two Hausdorff smooth manifolds is induced by a unique diffeomorphism between the manifolds, even without assuming second countability or paracompactness.
• The proof avoids reliance on multiplicative functionals by using a distinguished class of one-codimensional ideals corresponding to points via compactly supported smooth functions.
• A paracompact smooth manifold is smoothly realcompact if and only if the cardinality of its set of connected components is not $\aleph$-measurable.
• The standard proof of the isomorphism theorem fails for manifolds with $\aleph$-measurable component cardinalities because such manifolds admit free one-codimensional ideals.
• The method extends to $C^k$ manifolds and certain infinite-dimensional manifolds (e.g., modeled on Hilbert or $L^p$ spaces) if smooth bump functions with point support exist.
• A new proof of the Gel'fand-Kolmogoroff theorem is obtained without using the Stone-Čech compactification, by directly constructing the homeomorphism via distinguished ideals.