Theorems · Theorem · functional analysis

SpectrumRestricts.cfc_eq_restrict

∀ {R : Type u_1} {S : Type u_2} {A : Type u_3} {p q : A → Prop} [inst : Semifield R] [inst_1 : StarRing R]
  [inst_2 : MetricSpace R] [inst_3 : IsTopologicalSemiring R] [inst_4 : ContinuousStar R] [inst_5 : Semifield S]
  [inst_6 : StarRing S] [inst_7 : MetricSpace S] [inst_8 : IsTopologicalSemiring S] [inst_9 : ContinuousStar S]
  [inst_10 : Ring A] [inst_11 : StarRing A] [inst_12 : Algebra S A] [inst_13 : Algebra R S] [inst_14 : Algebra R A]
  [IsScalarTower R S A] [StarModule R S] [ContinuousSMul R S] [inst_18 : TopologicalSpace A]
  [inst_19 : ContinuousFunctionalCalculus S A q] [inst_20 : ContinuousFunctionalCalculus R A p]
  [ContinuousMap.UniqueHom R A] (f : C(S, R)),
  Topology.IsClosedEmbedding ⇑(algebraMap R S) →
    ∀ {a : A}, p a → q a → SpectrumRestricts a ⇑f → ∀ (g : R → R), cfc g a = cfc (fun x => (algebraMap R S) (g (f x))) a

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict

Cited by

2 results in Mathlib

Foundations

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

Assumes

SemifieldStarRingMetricSpaceIsTopologicalSemiringContinuousStarSemifieldStarRingMetricSpaceIsTopologicalSemiringContinuousStarRingStarRingAlgebraAlgebraAlgebraIsScalarTowerStarModuleContinuousSMulTopologicalSpaceContinuousFunctionalCalculusContinuousFunctionalCalculusContinuousMap.UniqueHom