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An Application of Descriptive Set Theory to Complex Analysis

Christopher Caruvana, Robert R. Kallman

Abstract

The purpose of this paper is to prove a new general result about rings of complex analytic functions. Let $Ω$ be an arbitrary nonempty open subset of the complex plane $\mathbb C$, $\mathcal{A}(Ω)$ be the set of holomorphic functions on $Ω$ viewed as a Polish ring (not a Polish algebra over $\mathbb C$) in the usual compact open topology, let $R$ be a Polish ring and let $\varphi : R \to \mathcal{A}(Ω)$ be an abstract algebraic isomorphism. The main goal of this paper is to prove Theorem 36 that $\varphi$ is a topological isomorphism. A special result of Bers is an easy corollary. Two additional items supplement these results, viz., that $B(\mathbb{D})$, the abstract ring of bounded analytic functions on the unit disk, cannot be made into a Polish ring and that $\mathcal{M}(Ω)$, the abstract field of meromorphic functions on $Ω$, cannot be made into a Polish field.

An Application of Descriptive Set Theory to Complex Analysis

Abstract

The purpose of this paper is to prove a new general result about rings of complex analytic functions. Let be an arbitrary nonempty open subset of the complex plane , be the set of holomorphic functions on viewed as a Polish ring (not a Polish algebra over ) in the usual compact open topology, let be a Polish ring and let be an abstract algebraic isomorphism. The main goal of this paper is to prove Theorem 36 that is a topological isomorphism. A special result of Bers is an easy corollary. Two additional items supplement these results, viz., that , the abstract ring of bounded analytic functions on the unit disk, cannot be made into a Polish ring and that , the abstract field of meromorphic functions on , cannot be made into a Polish field.

Paper Structure

This paper contains 8 sections, 45 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Borel Set Preliminaries
  3. General Polish Ring Preliminaries
  4. The Ring of Analytic Functions and Special Cases
  5. The General Case for the Ring of Analytic Functions
  6. A Theorem of Bers
  7. The Bounded Analytic Functions on the Disk
  8. The Field of Meromorphic Functions

Key Result

Theorem 1

Let $\varphi: G \to H$ be a Baire Property measurable homomorphism between Polish groups. Then $\varphi$ is continuous. If moreover, $\varphi[G]$ is not meager, then $\varphi$ is also open.

Theorems & Definitions (76)

  • Theorem 1: becker-kechris-1996a
  • Theorem 2: kechris-1995a
  • Proposition 3: atim-kallman-2012
  • Corollary 4
  • Theorem 5: Mackey mackey-1957a
  • Corollary 6
  • proof
  • Corollary 7
  • proof
  • Lemma 8
  • ...and 66 more