`BookProof.ChapterAbelianCyclicCommutant.denseRange_repVec_gelfandRep` {rho : A →⋆ₐ[ℂ] (H →L[ℂ] H)} {xi : H} (hcyc : DenseRange fun a : A => rho a xi) : DenseRange (repVec (gelfand

Proved

BookProof.ChapterAbelianCyclicCommutant.denseRange_repVec_gelfandRep

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAbelianCyclicCommutant.

BookProof.ChapterAbelianCyclicCommutant.denseRange_repVec_gelfandRep {rho : A →⋆ₐ[ℂ] (H →L[ℂ] H)} {xi : H} (hcyc : DenseRange fun a : A => rho a xi) : DenseRange (repVec (gelfandRep rho) xi)

Formalization note: Lean 4 identifier BookProof.ChapterAbelianCyclicCommutant.denseRange_repVec_gelfandRep.

Preamble

-- Generated from ChapterAbelianCyclicCommutant.lean — theorem BookProof.ChapterAbelianCyclicCommutant.denseRange_repVec_gelfandRep
import Definitions.Def_ChapterLinftyMultiplication
import Definitions.Def_ChapterLinftyMaximalAbelian
import Definitions.Def_ChapterAbelianGelfandModel
import Definitions.Def_ChapterSpectralCommutant
import Definitions.Def_ChapterAbelianCyclicModel
import Mathlib
import Definitions.Def_ChapterAbelianCyclicCommutant
open BookProof.ChapterAbelianCyclicCommutant


noncomputable section

open MeasureTheory Complex WeakDual


open BookProof.ChapterLinftyMultiplication BookProof.ChapterLinftyMaximalAbelian
open BookProof.ChapterAbelianGelfandModel BookProof.ChapterSpectralCommutant
open 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)) (xi : H) (hcyc : DenseRange (repVec pi xi))

variable {A : Type*} [CommCStarAlgebra A]

Formal statement

theorem BookProof.ChapterAbelianCyclicCommutant.denseRange_repVec_gelfandRep {rho : A →⋆ₐ[ℂ] (H →L[ℂ] H)} {xi : H}
    (hcyc : DenseRange fun a : A => rho a xi) :
    DenseRange (repVec (gelfandRep rho) xi) := by sorry

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