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Evaluative presentations

Timothy H. McNicholl

Abstract

We study presentations of $C^*(X)$ that are evaluative over a presentation of $X$ in that $(f,p) \mapsto f(p)$ is computable. We prove existence-uniqueness theorems for such presentations. We use our methods to prove an effective Banach-Stone Theorem for unital commutative $C^*$ algebras. We also apply our results to the computable categoricity of $C^*$ algebras and compact Polish spaces.

Evaluative presentations

Abstract

We study presentations of that are evaluative over a presentation of in that is computable. We prove existence-uniqueness theorems for such presentations. We use our methods to prove an effective Banach-Stone Theorem for unital commutative algebras. We also apply our results to the computable categoricity of algebras and compact Polish spaces.
Paper Structure (11 sections, 23 theorems, 2 equations)

This paper contains 11 sections, 23 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Preliminaries
  4. Proofs of the main theorems
  5. Uniformity and non-uniformity results
  6. Applications
  7. Computing spatial realizations and composition operators
  8. Computable categoricity and related topics
  9. The classification problem for unital commutative $C^*$ algebras
  10. Conclusion
  11. Acknowledgement

Key Result

Theorem 1.1

Suppose $X$ is a compact Polish space. If $C^*(X)^\#$ is a computable presentation, then, up to computable homeomorphism, there is a unique presentation of $X$ over which $C^*(X)^\#$ is evaluative.

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Theorem 2.2
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 26 more