Theorems · Theorem · functional analysis

algebraMap_le_cfc

∀ {R : Type u_1} {A : Type u_2} {p : A → Prop} [inst : CommSemiring R] [inst_1 : PartialOrder R] [inst_2 : StarRing R]
  [inst_3 : MetricSpace R] [inst_4 : IsTopologicalSemiring R] [inst_5 : ContinuousStar R] [ContinuousSqrt R]
  [StarOrderedRing R] [inst_8 : TopologicalSpace A] [inst_9 : Ring A] [inst_10 : StarRing A] [inst_11 : PartialOrder A]
  [StarOrderedRing A] [inst_13 : Algebra R A] [instCFC : ContinuousFunctionalCalculus R A p] (f : R → R) (r : R)
  (a : A),
  (∀ x ∈ spectrum R a, r ≤ f x) →
    autoParam (ContinuousOn f (spectrum R a)) algebraMap_le_cfc._auto_1 →
      autoParam (p a) algebraMap_le_cfc._auto_3 → (algebraMap R A) r ≤ cfc f a

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Cited by

2 results in Mathlib

Foundations

Depth 102 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringPartialOrderStarRingMetricSpaceIsTopologicalSemiringContinuousStarContinuousSqrtStarOrderedRingTopologicalSpaceRingStarRingPartialOrderStarOrderedRingAlgebraContinuousFunctionalCalculus