`BookProof.ChapterAbelianCyclicModel.abelian_algebra_multiplication_model` (rho : A →⋆ₐ[ℂ] (H →L[ℂ] H)) (xi : H) (hxi : ‖xi‖ = 1) (hcyc : DenseRange fun a : A => rho a xi) : ∃ (mu

Proved

BookProof.ChapterAbelianCyclicModel.abelian_algebra_multiplication_model

by leonardopedro · Oct 17, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAbelianCyclicModel.

BookProof.ChapterAbelianCyclicModel.abelian_algebra_multiplication_model (rho : A →⋆ₐ[ℂ] (H →L[ℂ] H)) (xi : H) (hxi : ‖xi‖ = 1) (hcyc : DenseRange fun a : A => rho a xi) : ∃ (mu : Measure (characterSpace ℂ A)) (_ : IsProbabilityMeasure mu) (U : Lp ℂ 2 mu ≃ₗᵢ[ℂ] H), ∀ (a : A) (u : Lp ℂ 2 mu), U (mulRep mu (gelfandModel A a) u) = rho a (U u)

Formalization note: Lean 4 identifier BookProof.ChapterAbelianCyclicModel.abelian_algebra_multiplication_model.

Preamble

-- Generated from ChapterAbelianCyclicModel.lean — theorem BookProof.ChapterAbelianCyclicModel.abelian_algebra_multiplication_model
import Definitions.Def_ChapterAbelianGelfandModel
import Mathlib
import Definitions.Def_ChapterAbelianCyclicModel
open BookProof.ChapterAbelianCyclicModel


open MeasureTheory Complex WeakDual
open scoped ComplexOrder


open BookProof.ChapterAbelianGelfandModel


variable {X : Type*} [TopologicalSpace X] [CompactSpace X] [T2Space X]
  [MeasurableSpace X] [BorelSpace X]
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
variable (pi : C(X, ℂ) →⋆ₐ[ℂ] (H →L[ℂ] H)) (xi : H)

variable (hcyc : DenseRange (repVec pi xi))
variable {A : Type*} [CommCStarAlgebra A]

Formal statement

theorem BookProof.ChapterAbelianCyclicModel.abelian_algebra_multiplication_model (rho : A →⋆ₐ[ℂ] (H →L[ℂ] H)) (xi : H)
    (hxi : ‖xi‖ = 1) (hcyc : DenseRange fun a : A => rho a xi) :
    ∃ (mu : Measure (characterSpace ℂ A)) (_ : IsProbabilityMeasure mu)
      (U : Lp ℂ 2 mu ≃ₗᵢ[ℂ] H),
      ∀ (a : A) (u : Lp ℂ 2 mu), U (mulRep mu (gelfandModel A a) u) = rho a (U u) := by sorry

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