`BookProof.ChapterAbelianGelfandModel.state_is_vector_state_of_multiplication` (phi : A →ₗ[ℂ] ℂ) (hpos : ∀ a : A, 0 ≤ phi (star a * a)) (hone : phi 1 = 1) : ∃ mu : Measure (charact

Proved

BookProof.ChapterAbelianGelfandModel.state_is_vector_state_of_multiplication

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAbelianGelfandModel.

BookProof.ChapterAbelianGelfandModel.state_is_vector_state_of_multiplication (phi : A →ₗ[ℂ] ℂ) (hpos : ∀ a : A, 0 ≤ phi (star a * a)) (hone : phi 1 = 1) : ∃ mu : Measure (characterSpace ℂ A), ∃ _ : IsProbabilityMeasure mu, ‖oneVec mu‖ = 1 ∧ ∀ a : A, phi a = inner ℂ (oneVec mu) (multiplicationRep mu a (oneVec mu))

Formalization note: Lean 4 identifier BookProof.ChapterAbelianGelfandModel.state_is_vector_state_of_multiplication.

Preamble

-- Generated from ChapterAbelianGelfandModel.lean — theorem BookProof.ChapterAbelianGelfandModel.state_is_vector_state_of_multiplication
import Definitions.Def_ChapterLinftyMultiplication
import Mathlib
import Definitions.Def_ChapterAbelianGelfandModel
open BookProof.ChapterAbelianGelfandModel


open MeasureTheory Complex WeakDual CompactlySupported CompactlySupportedContinuousMap
open scoped ComplexOrder


open BookProof.ChapterLinftyMultiplication

variable {X : Type*} [TopologicalSpace X] [CompactSpace X] [T2Space X]
  [MeasurableSpace X] [BorelSpace X]
variable {X : Type*} [TopologicalSpace X]
variable {X : Type*} [TopologicalSpace X] [CompactSpace X] [T2Space X]
  [MeasurableSpace X] [BorelSpace X]
  (psi : C(X, ℂ) →ₗ[ℂ] ℂ) (hpos : ∀ g : C(X, ℂ), 0 ≤ psi (star g * g))
variable {X : Type*} [MeasurableSpace X] (mu : Measure X)
variable {X : Type*} [TopologicalSpace X] [CompactSpace X]
  [MeasurableSpace X] [BorelSpace X] (mu : Measure X)
variable (A : Type*) [CommCStarAlgebra A]
variable {A}

Formal statement

theorem BookProof.ChapterAbelianGelfandModel.state_is_vector_state_of_multiplication (phi : A →ₗ[ℂ] ℂ)
    (hpos : ∀ a : A, 0 ≤ phi (star a * a)) (hone : phi 1 = 1) :
    ∃ mu : Measure (characterSpace ℂ A), ∃ _ : IsProbabilityMeasure mu,
      ‖oneVec mu‖ = 1 ∧
      ∀ a : A, phi a = inner ℂ (oneVec mu) (multiplicationRep mu a (oneVec mu)) := by sorry

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