Theorems · Theorem · functional analysis

ContinuousAt.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 : Set NNReal},
  IsCompact s →
    ∀ (f : NNReal → NNReal) {a : X → A} {x₀ : X},
      ContinuousAt a x₀ →
        (∀ᶠ (x : X) in nhds x₀, spectrum NNReal (a x) ⊆ s) →
          (∀ᶠ (x : X) in nhds x₀, 0 ≤ a x) →
            autoParam (ContinuousOn f s) ContinuousAt.cfc_nnreal._auto_1 → ContinuousAt (fun x => cfc f (a x)) x₀

If f : ℝ≥0 → ℝ≥0 is continuous on a compact set s and a : X → A is continuous at x₀, and eventually 0 ≤ a x and has spectrum contained in s, then fun x ↦ cfc f (a x) is continuous at x₀.

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Cited by

0 results in Mathlib

Foundations

Depth 201 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedRingStarRingNormedAlgebraIsometricContinuousFunctionalCalculusContinuousStarPartialOrderStarOrderedRingNonnegSpectrumClassT2SpaceIsSemitopologicalRingTopologicalSpace