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[Merged by Bors] - doc(Analysis/NormedSpace/Star/GelfandDuality): update TODO list #10763

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[Merged by Bors] - doc(Analysis/NormedSpace/Star/GelfandDuality): update TODO list #10763

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14 changes: 4 additions & 10 deletions Mathlib/Analysis/NormedSpace/Star/GelfandDuality.lean
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Expand Up @@ -52,16 +52,10 @@ Then `η₁ : id → F ∘ G := gelfandStarTransform` and

## TODO

* After `StarAlgEquiv` is defined, realize `gelfandTransform` as a `StarAlgEquiv`.
* Prove that if `A` is the unital C⋆-algebra over `ℂ` generated by a fixed normal element `x` in
a larger C⋆-algebra `B`, then `characterSpace ℂ A` is homeomorphic to `spectrum ℂ x`.
* From the previous result, construct the **continuous functional calculus**.
* Show that if `X` is a compact Hausdorff space, then `X` is (canonically) homeomorphic to
`characterSpace ℂ C(X, ℂ)`.
* Conclude using the previous fact that the functors `C(·, ℂ)` and `characterSpace ℂ ·` along with
the canonical homeomorphisms described above constitute a natural contravariant equivalence of
the categories of compact Hausdorff spaces (with continuous maps) and commutative unital
C⋆-algebras (with unital ⋆-algebra homomorphisms); this is known as **Gelfand duality**.
* After defining the category of commutative unital C⋆-algebras, bundle the existing unbundled
**Gelfand duality** into an actual equivalence (duality) of categories associated to the
functors `C(·, ℂ)` and `characterSpace ℂ ·` and the natural isomorphisms `gelfandStarTransform`
and `WeakDual.CharacterSpace.homeoEval`.

## Tags

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