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Computable Gelfand Duality

Peter Burton, Christopher J. Eagle, Alec Fox, Isaac Goldbring, Matthew Harrison-Trainor, Timothy H. McNicholl, Alexander Melnikov, Teerawat Thewmorakot

Abstract

We establish a computable version of Gelfand Duality. Under this computable duality, computably compact presentations of metrizable spaces uniformly effectively correspond to computable presentations of unital commutative $C^*$ algebras.

Computable Gelfand Duality

Abstract

We establish a computable version of Gelfand Duality. Under this computable duality, computably compact presentations of metrizable spaces uniformly effectively correspond to computable presentations of unital commutative algebras.
Paper Structure (7 sections, 16 theorems, 12 equations)

This paper contains 7 sections, 16 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Preliminaries from classical analysis
  4. Computability-theoretic preliminaries
  5. Proof of Theorem \ref{['thm:main']}
  6. Applications and extensions
  7. Conclusion

Key Result

Theorem 1.1

Suppose $X$ is a compact metrizable space. If $C^*(X)$ is computably presentable, then $X$ has a computably compact presentation.

Theorems & Definitions (25)

  • Theorem 1.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 4.1
  • proof
  • Proposition 4.2
  • ...and 15 more