Theorems · Theorem · functional analysis

cfc_nnreal_le_iff

∀ {A : Type u_1} [inst : TopologicalSpace A] [inst_1 : Ring A] [inst_2 : StarRing A] [inst_3 : PartialOrder A]
  [inst_4 : StarOrderedRing A] [inst_5 : Algebra ℝ A] [IsTopologicalRing A] [inst_7 : NonnegSpectrumClass ℝ A]
  [T2Space A] [inst_9 : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint] (f g : NNReal → NNReal) (a : A),
  SpectrumRestricts a ⇑ContinuousMap.realToNNReal →
    autoParam (ContinuousOn f (spectrum NNReal a)) cfc_nnreal_le_iff._auto_1 →
      autoParam (ContinuousOn g (spectrum NNReal a)) cfc_nnreal_le_iff._auto_3 →
        autoParam (0 ≤ a) cfc_nnreal_le_iff._auto_5 → (cfc f a ≤ cfc g a ↔ ∀ x ∈ spectrum NNReal a, f x ≤ g x)

cfc_le_iff only applies to a scalar ring where R is an actual Ring, and not a Semiring. However, this theorem still holds for ℝ≥0 as long as the algebra A itself is an ℝ-algebra.

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order

Cited by

0 results in Mathlib

Foundations

Depth 199 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceRingStarRingPartialOrderStarOrderedRingAlgebraIsTopologicalRingNonnegSpectrumClassT2SpaceContinuousFunctionalCalculus