`BookProof.ChapterAbelianDirectSum.abelian_algebra_multiplication_model_general` (rho : A →⋆ₐ[ℂ] (H →L[ℂ] H)) : ∃ (S : Set H) (mu : S → Measure (characterSpace ℂ A)) (V : ∀ x : S,

Proved

BookProof.ChapterAbelianDirectSum.abelian_algebra_multiplication_model_general

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAbelianDirectSum.

BookProof.ChapterAbelianDirectSum.abelian_algebra_multiplication_model_general (rho : A →⋆ₐ[ℂ] (H →L[ℂ] H)) : ∃ (S : Set H) (mu : S → Measure (characterSpace ℂ A)) (V : ∀ x : S, Lp ℂ 2 (mu x) →ₗᵢ[ℂ] H), (∀ x : S, IsProbabilityMeasure (mu x)) ∧ IsHilbertSum ℂ (fun x : S => Lp ℂ 2 (mu x)) V ∧ (∀ (x : S) (a : A) (u : Lp ℂ 2 (mu x)), V x (mulRep (mu x) (gelfandModel A a) u) = rho a (V x u))

Formalization note: Lean 4 identifier BookProof.ChapterAbelianDirectSum.abelian_algebra_multiplication_model_general.

Preamble

-- Generated from ChapterAbelianDirectSum.lean — theorem BookProof.ChapterAbelianDirectSum.abelian_algebra_multiplication_model_general
import Definitions.Def_ChapterAbelianGelfandModel
import Definitions.Def_ChapterAbelianCyclicModel
import Mathlib
import Definitions.Def_ChapterAbelianDirectSum
open BookProof.ChapterAbelianDirectSum


noncomputable section

open MeasureTheory Complex WeakDual


open BookProof.ChapterAbelianGelfandModel BookProof.ChapterAbelianCyclicModel

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))

variable (xi : H)
variable {pi}
variable (pi)
variable {A : Type*} [CommCStarAlgebra A]

Formal statement

theorem BookProof.ChapterAbelianDirectSum.abelian_algebra_multiplication_model_general (rho : A →⋆ₐ[ℂ] (H →L[ℂ] H)) :
    ∃ (S : Set H) (mu : S → Measure (characterSpace ℂ A))
      (V : ∀ x : S, Lp ℂ 2 (mu x) →ₗᵢ[ℂ] H),
      (∀ x : S, IsProbabilityMeasure (mu x)) ∧
      IsHilbertSum ℂ (fun x : S => Lp ℂ 2 (mu x)) V ∧
      (∀ (x : S) (a : A) (u : Lp ℂ 2 (mu x)),
        V x (mulRep (mu x) (gelfandModel A a) u) = rho a (V x u)) := by sorry

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