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Range decreasing group homomorphisms and weighted composition operators

Ning Zhang

TL;DR

The paper develops a unifying framework to recognize weighted composition operators as the canonical form of many group-, semigroup-, Lie algebra-, and algebra-homomorphisms between spaces of smooth sections of bundles. The core method, range decreasing group homomorphisms, yields necessary and sufficient criteria that force such maps to be weighted composition operators in broad settings, often reducing to an evaluation at a point and a pullback along a base-map $\phi$. Across multiple layers—mapping groups, linear maps, semigroups, bundle sections, vector fields, and algebras—the authors show that the algebraic structure of the space of sections largely determines the underlying bundle and its morphisms, with automatic regularity (continuity and smoothness) under mild hypotheses. They derive a homomorphism version of the Shanks–Pursell theorem and extend Milgram–Mrčun–Šemrl-type results to wider contexts, including arbitrary finite-dimensional manifolds and infinite-dimensional fibers. The results illuminate how bundle geometry is encoded in section algebras and offer a robust toolkit for identifying weighted composition operators in geometric and algebraic settings.

Abstract

We present necessary and sufficient conditions for a group homomorphism between spaces of smooth sections of Lie group bundles to be a weighted composition operator. These results provide new insights into a wide range of problems related to weighted composition operators. Specifically, we prove that the algebraic structure of the space of smooth sections of an algebra bundle, where the typical fiber is a positive dimensional simple unital algebra, completely determines the bundle structure. Furthermore, we derive a homomorphism version of the Shanks-Pursell theorem and identify a class of homomorphisms of multiplicative semigroups between spaces of smooth functions on finite dimensional manifolds, including all isomorphisms. Our approach is based on a method called range decreasing group homomorphisms.

Range decreasing group homomorphisms and weighted composition operators

TL;DR

The paper develops a unifying framework to recognize weighted composition operators as the canonical form of many group-, semigroup-, Lie algebra-, and algebra-homomorphisms between spaces of smooth sections of bundles. The core method, range decreasing group homomorphisms, yields necessary and sufficient criteria that force such maps to be weighted composition operators in broad settings, often reducing to an evaluation at a point and a pullback along a base-map . Across multiple layers—mapping groups, linear maps, semigroups, bundle sections, vector fields, and algebras—the authors show that the algebraic structure of the space of sections largely determines the underlying bundle and its morphisms, with automatic regularity (continuity and smoothness) under mild hypotheses. They derive a homomorphism version of the Shanks–Pursell theorem and extend Milgram–Mrčun–Šemrl-type results to wider contexts, including arbitrary finite-dimensional manifolds and infinite-dimensional fibers. The results illuminate how bundle geometry is encoded in section algebras and offer a robust toolkit for identifying weighted composition operators in geometric and algebraic settings.

Abstract

We present necessary and sufficient conditions for a group homomorphism between spaces of smooth sections of Lie group bundles to be a weighted composition operator. These results provide new insights into a wide range of problems related to weighted composition operators. Specifically, we prove that the algebraic structure of the space of smooth sections of an algebra bundle, where the typical fiber is a positive dimensional simple unital algebra, completely determines the bundle structure. Furthermore, we derive a homomorphism version of the Shanks-Pursell theorem and identify a class of homomorphisms of multiplicative semigroups between spaces of smooth functions on finite dimensional manifolds, including all isomorphisms. Our approach is based on a method called range decreasing group homomorphisms.
Paper Structure (12 sections, 37 theorems, 76 equations)

This paper contains 12 sections, 37 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Range decreasing homomorphisms $\mathcal{F}_{c}\left(V, G \right) \to G$
  4. Range decreasing additive maps
  5. The non-surjective elements of $\mathcal{F}\left(V, G \right)$
  6. Automatic continuity and automatic smoothness
  7. Homomorphisms of mapping groups
  8. Linear maps between function spaces
  9. Homomorphisms of semigroups of functions
  10. Homomorphisms of groups of sections of bundles
  11. Homomorphisms of Lie algebras of vector fields
  12. Homomorphisms of algebras of sections of bundles

Key Result

Proposition 2.1

Let $f: \mathcal{F}\left(V, G \right) \to \mathcal{M}\left(S, G \right)$ be a group homomorphism such that $f(a)=a$ for each $a \in G$ and $f\left( \mathcal{F}(V, G \setminus \{\mathbf{1}\}) \right) \subset \mathcal{M}(S, G \setminus \{\mathbf{1}\})$. Then $f$ is range decreasing.

Theorems & Definitions (70)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 3.1
  • proof
  • ...and 60 more