Theorems · Theorem · functional analysis

Continuous.cfc_nnreal

∀ {X : Type u_1} {A : Type u_2} [inst : NormedRing A] [inst_1 : StarRing A] [inst_2 : NormedAlgebra ℝ A]
  [inst_3 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint] [ContinuousStar A] [inst_5 : PartialOrder A]
  [inst_6 : StarOrderedRing A] [inst_7 : NonnegSpectrumClass ℝ A] [T2Space A] [IsSemitopologicalRing A]
  [inst_10 : TopologicalSpace X] {s : X → Set NNReal} (f : NNReal → NNReal) {a : X → A},
  Continuous a →
    (∀ (x : X), IsCompact (s x)) →
      (∀ (x₀ : X), ∀ᶠ (x : X) in nhds x₀, spectrum NNReal (a x) ⊆ s x₀) →
        autoParam (∀ (x : X), ContinuousOn f (s x)) Continuous.cfc_nnreal._auto_1 →
          autoParam (∀ (x : X), 0 ≤ a x) Continuous.cfc_nnreal._auto_3 → Continuous fun x => cfc f (a x)

Suppose a : X → Set A is a continuous family of nonnegative elements. Suppose further that s : X → Set ℝ≥0 is a family of compact sets such that s x₀ contains the spectrum of a x for all sufficiently close x. If f : ℝ≥0 → ℝ≥0 is continuous on each s x, then fun x ↦ cfc f (a x) is continuous.

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Cited by

0 results in Mathlib

Foundations

Depth 203 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedRingStarRingNormedAlgebraIsometricContinuousFunctionalCalculusContinuousStarPartialOrderStarOrderedRingNonnegSpectrumClassT2SpaceIsSemitopologicalRingTopologicalSpace