OAI/Analysis/KadisonKastler/SpectralVectorMeasure.lean

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8 declarations, 0 contain sorry, 0 declared as axioms

Finite vector measures for real scalar multipliers

The squared pointwise norm of an L2 vector defines a finite measure. Character commutation preserves its characteristic function and hence the measure itself. This yields commutation with every scalar multiplier.

  1. def OAI.KadisonKastler.regularVectorMeasure line 21
  2. instance OAI.KadisonKastler.regularVectorMeasure_finite line 24
  3. theorem OAI.KadisonKastler.regularVectorMeasure_integral line 32
  4. theorem OAI.KadisonKastler.regularScalarMultiplier_apply_ae line 41
  5. theorem OAI.KadisonKastler.regularVectorMeasure_multiplier line 51
  6. theorem OAI.KadisonKastler.regularVectorMeasure_charFun line 63
  7. theorem OAI.KadisonKastler.regularVectorMeasure_unitary_eq line 74
  8. theorem OAI.KadisonKastler.unitary_commute_scalarMultiplier_of_characters line 95

    Source documentation

    Invariance of all vector measures implies commutation with every bounded Borel multiplier, by polarization of its quadratic form.

    theorem unitary_commute_scalarMultiplier_of_characters
    (U : unitary (RegularCoreHilbert H →L[ℂ] RegularCoreHilbert H))
    (hU : ∀ a : ℝ, Commute (U : RegularCoreHilbert H →L[ℂ] RegularCoreHilbert H)
      (regularCharacterUnitary (H := H) a : RegularCoreHilbert H →L[ℂ] RegularCoreHilbert H))
…

    Mentions: OAI.KadisonKastler.RegularCoreHilbert, OAI.KadisonKastler.regularCharacterUnitary