feat: naturality of the gelfandStarTransform and `WeakDual.Characte…

…rSpace.homeoEval` (#9638)

Mathlib/Analysis/NormedSpace/Star/GelfandDuality.lean

@@ -9,6 +9,7 @@ import Mathlib.Analysis.NormedSpace.Algebra
 import Mathlib.Topology.ContinuousFunction.Units
 import Mathlib.Topology.ContinuousFunction.Compact
 import Mathlib.Topology.Algebra.Algebra
+import Mathlib.Topology.ContinuousFunction.Ideals
 import Mathlib.Topology.ContinuousFunction.StoneWeierstrass
 
 #align_import analysis.normed_space.star.gelfand_duality from "leanprover-community/mathlib"@"e65771194f9e923a70dfb49b6ca7be6e400d8b6f"
@@ -22,12 +23,21 @@ the Gelfand transform is actually spectrum-preserving (`spectrum.gelfandTransfor
 when `A` is a commutative C⋆-algebra over `ℂ`, then the Gelfand transform is a surjective isometry,
 and even an equivalence between C⋆-algebras.
 
+Consider the contravariant functors between compact Hausdorff spaces and commutative unital
+C⋆algebras `F : Cpct → CommCStarAlg := X ↦ C(X, ℂ)` and
+`G : CommCStarAlg → Cpct := A → characterSpace ℂ A` whose actions on morphisms are given by
+`WeakDual.CharacterSpace.compContinuousMap` and `ContinuousMap.compStarAlgHom'`, respectively.
+
+Then `η₁ : id → F ∘ G := gelfandStarTransform` and
+`η₂ : id → G ∘ F := WeakDual.CharacterSpace.homeoEval` are the natural isomorphisms implementing
+**Gelfand Duality**, i.e., the (contravariant) equivalence of these categories.
+
 ## Main definitions
 
 * `Ideal.toCharacterSpace` : constructs an element of the character space from a maximal ideal in
   a commutative complex Banach algebra
-* `WeakDual.CharacterSpace.compContinuousMap`: The functorial map taking `ψ : A →⋆ₐ[ℂ] B` to a
-  continuous function `characterSpace ℂ B → characterSpace ℂ A` given by pre-composition with `ψ`.
+* `WeakDual.CharacterSpace.compContinuousMap`: The functorial map taking `ψ : A →⋆ₐ[𝕜] B` to a
+  continuous function `characterSpace 𝕜 B → characterSpace 𝕜 A` given by pre-composition with `ψ`.
 
 ## Main statements
 
@@ -37,6 +47,8 @@ and even an equivalence between C⋆-algebras.
   commutative (unital) C⋆-algebra over `ℂ`.
 * `gelfandTransform_bijective` : the Gelfand transform is bijective when the algebra is a
   commutative (unital) C⋆-algebra over `ℂ`.
+* `gelfandStarTransform_naturality`: The `gelfandStarTransform` is a natural isomorphism
+* `WeakDual.CharacterSpace.homeoEval_naturality`: This map implements a natural isomorphism
 
 ## TODO
 
@@ -206,39 +218,36 @@ namespace WeakDual
 
 namespace CharacterSpace
 
-variable {A B C : Type*}
-
-variable [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [StarRing A]
-
-variable [NormedRing B] [NormedAlgebra ℂ B] [CompleteSpace B] [StarRing B]
-
-variable [NormedRing C] [NormedAlgebra ℂ C] [CompleteSpace C] [StarRing C]
+variable {A B C 𝕜 : Type*} [NontriviallyNormedField 𝕜]
+variable [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] [StarRing A]
+variable [NormedRing B] [NormedAlgebra 𝕜 B] [CompleteSpace B] [StarRing B]
+variable [NormedRing C] [NormedAlgebra 𝕜 C] [CompleteSpace C] [StarRing C]
 
 /-- The functorial map taking `ψ : A →⋆ₐ[ℂ] B` to a continuous function
 `characterSpace ℂ B → characterSpace ℂ A` obtained by pre-composition with `ψ`. -/
 @[simps]
-noncomputable def compContinuousMap (ψ : A →⋆ₐ[ℂ] B) : C(characterSpace ℂ B, characterSpace ℂ A)
+noncomputable def compContinuousMap (ψ : A →⋆ₐ[𝕜] B) : C(characterSpace 𝕜 B, characterSpace 𝕜 A)
     where
   toFun φ := equivAlgHom.symm ((equivAlgHom φ).comp ψ.toAlgHom)
   continuous_toFun :=
     Continuous.subtype_mk
-      (continuous_of_continuous_eval fun a => map_continuous <| gelfandTransform ℂ B (ψ a)) _
+      (continuous_of_continuous_eval fun a => map_continuous <| gelfandTransform 𝕜 B (ψ a)) _
 #align weak_dual.character_space.comp_continuous_map WeakDual.CharacterSpace.compContinuousMap
 
 variable (A)
 
 /-- `WeakDual.CharacterSpace.compContinuousMap` sends the identity to the identity. -/
 @[simp]
 theorem compContinuousMap_id :
-    compContinuousMap (StarAlgHom.id ℂ A) = ContinuousMap.id (characterSpace ℂ A) :=
+    compContinuousMap (StarAlgHom.id 𝕜 A) = ContinuousMap.id (characterSpace 𝕜 A) :=
   ContinuousMap.ext fun _a => ext fun _x => rfl
 #align weak_dual.character_space.comp_continuous_map_id WeakDual.CharacterSpace.compContinuousMap_id
 
 variable {A}
 
 /-- `WeakDual.CharacterSpace.compContinuousMap` is functorial. -/
 @[simp]
-theorem compContinuousMap_comp (ψ₂ : B →⋆ₐ[ℂ] C) (ψ₁ : A →⋆ₐ[ℂ] B) :
+theorem compContinuousMap_comp (ψ₂ : B →⋆ₐ[𝕜] C) (ψ₁ : A →⋆ₐ[𝕜] B) :
     compContinuousMap (ψ₂.comp ψ₁) = (compContinuousMap ψ₁).comp (compContinuousMap ψ₂) :=
   ContinuousMap.ext fun _a => ext fun _x => rfl
 #align weak_dual.character_space.comp_continuous_map_comp WeakDual.CharacterSpace.compContinuousMap_comp
@@ -248,3 +257,64 @@ end CharacterSpace
 end WeakDual
 
 end Functoriality
+
+open CharacterSpace in
+/--
+Consider the contravariant functors between compact Hausdorff spaces and commutative unital
+C⋆algebras `F : Cpct → CommCStarAlg := X ↦ C(X, ℂ)` and
+`G : CommCStarAlg → Cpct := A → characterSpace ℂ A` whose actions on morphisms are given by
+`WeakDual.CharacterSpace.compContinuousMap` and `ContinuousMap.compStarAlgHom'`, respectively.
+
+Then `η : id → F ∘ G := gelfandStarTransform` is a natural isomorphism implementing (half of)
+the duality between these categories. That is, for commutative unital C⋆-algebras `A` and `B` and
+`φ : A →⋆ₐ[ℂ] B` the following diagram commutes:
+
+```
+A  --- η A ---> C(characterSpace ℂ A, ℂ)
+
+|                     |
+
+φ                  (F ∘ G) φ
+
+|                     |
+V                     V
+
+B  --- η B ---> C(characterSpace ℂ B, ℂ)
+```
+-/
+theorem gelfandStarTransform_naturality {A B : Type*} [NormedCommRing A] [NormedAlgebra ℂ A]
+    [CompleteSpace A] [StarRing A] [CstarRing A] [StarModule ℂ A] [NormedCommRing B]
+    [NormedAlgebra ℂ B] [CompleteSpace B] [StarRing B] [CstarRing B] [StarModule ℂ B]
+    (φ : A →⋆ₐ[ℂ] B) :
+    (gelfandStarTransform B : _ →⋆ₐ[ℂ] _).comp φ =
+      (compContinuousMap φ |>.compStarAlgHom' ℂ ℂ).comp (gelfandStarTransform A : _ →⋆ₐ[ℂ] _) := by
+  rfl
+
+/--
+Consider the contravariant functors between compact Hausdorff spaces and commutative unital
+C⋆algebras `F : Cpct → CommCStarAlg := X ↦ C(X, ℂ)` and
+`G : CommCStarAlg → Cpct := A → characterSpace ℂ A` whose actions on morphisms are given by
+`WeakDual.CharacterSpace.compContinuousMap` and `ContinuousMap.compStarAlgHom'`, respectively.
+
+Then `η : id → G ∘ F := WeakDual.CharacterSpace.homeoEval` is a natural isomorphism implementing
+(half of) the duality between these categories. That is, for compact Hausdorff spaces `X` and `Y`,
+`f : C(X, Y)` the following diagram commutes:
+
+```
+X  --- η X ---> characterSpace ℂ C(X, ℂ)
+
+|                     |
+
+f                  (G ∘ F) f
+
+|                     |
+V                     V
+
+Y  --- η Y ---> characterSpace ℂ C(Y, ℂ)
+```
+-/
+lemma WeakDual.CharacterSpace.homeoEval_naturality {X Y 𝕜 : Type*} [IsROrC 𝕜] [TopologicalSpace X]
+    [CompactSpace X] [T2Space X] [TopologicalSpace Y] [CompactSpace Y] [T2Space Y] (f : C(X, Y)) :
+    (homeoEval Y 𝕜 : C(_, _)).comp f =
+      (f.compStarAlgHom' 𝕜 𝕜 |> compContinuousMap).comp (homeoEval X 𝕜 : C(_, _)) :=
+  rfl