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Applications of the Gelfand--Naimark duality

Ilijas Farah

Abstract

Stone duality is an indispensable tool for the study of compact, zero-dimensional, Hausdorff spaces. In the case of general compact Hausdorff spaces one can get quite a bit of mileage by considering the `Wallman duality' between compact spaces and lattices of closed sets. I will argue that the Gelfand--Naimark duality between compact Hausdorff spaces and unital, commutative \cstar-algebras provides great insight into compact Hausdorff spaces, and \v Cech--Stone remainders and their autohomeomorphisms in particular.

Applications of the Gelfand--Naimark duality

Abstract

Stone duality is an indispensable tool for the study of compact, zero-dimensional, Hausdorff spaces. In the case of general compact Hausdorff spaces one can get quite a bit of mileage by considering the `Wallman duality' between compact spaces and lattices of closed sets. I will argue that the Gelfand--Naimark duality between compact Hausdorff spaces and unital, commutative \cstar-algebras provides great insight into compact Hausdorff spaces, and \v Cech--Stone remainders and their autohomeomorphisms in particular.
Paper Structure (19 sections, 24 theorems, 30 equations)

This paper contains 19 sections, 24 theorems, 30 equations.

Table of Contents

  1. Preliminaries
  2. Stone duality
  3. $\mathrm{C}^*$-algebras and Gelfand--Naimark duality
  4. Hilbert space
  5. Algebras
  6. Compact Hausdorff spaces
  7. Locally compact Hausdorff spaces
  8. Model theory
  9. Syntax and semantics
  10. Reduced coproducts, ultracoproducts, and cotheories
  11. Types and saturation
  12. Case study: Applications to continua
  13. Čech--Stone remainders
  14. Continuum Hypothesis
  15. Nontrivial autohomeomorphisms without countable saturation.
  16. ...and 4 more sections

Key Result

Theorem 1.1

The functor $X\mapsto \mathop{\mathrm{Clop}}\nolimits(X)$ is an equivalence of categories of compact, Hausdorff, zero-dimensional spaces and Boolean algebras. This functor is contravariant. More precisely, we have the following for every continuous function $f\colon X\to Y$ between compact, Hausdorf

Theorems & Definitions (53)

  • Theorem 1.1
  • Definition 1.2
  • Remark 1.3
  • Example 1.4
  • Definition 1.5
  • Example 1.6
  • Theorem 1.7
  • proof
  • Lemma 1.8
  • proof
  • ...and 43 more