| import Mathlib | |
| open CategoryTheory | |
| universe u | |
| structure CommCStarAlgCat : Type (u + 1) where | |
| of :: | |
| carrier : Type u | |
| [commCStarAlgebra : CommCStarAlgebra carrier] | |
| attribute [instance] CommCStarAlgCat.commCStarAlgebra | |
| namespace CommCStarAlgCat | |
| instance : CoeSort CommCStarAlgCat (Type u) := | |
| ⟨CommCStarAlgCat.carrier⟩ | |
| instance : Category CommCStarAlgCat where | |
| Hom A B := A →⋆ₐ[ℂ] B | |
| id A := StarAlgHom.id ℂ A | |
| comp f g := g.comp f | |
| end CommCStarAlgCat | |
| theorem gelfandDuality : Nonempty (CompHaus.{u} ≌ (CommCStarAlgCat.{u})ᵒᵖ) := by | |
| sorry | |