Theorems · Theorem · functional analysis

SpectrumRestricts.cfcHom_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]
  [inst_15 : IsScalarTower R S A] [inst_16 : StarModule R S] [inst_17 : 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)) {a : A} (hpa : p a)
  (hqa : q a) (h : SpectrumRestricts a ⇑f), cfcHom hpa = SpectrumRestricts.starAlgHom (cfcHom hqa) h

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict

Cited by

5 results in Mathlib

Foundations

Depth 98 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemifieldStarRingMetricSpaceIsTopologicalSemiringContinuousStarSemifieldStarRingMetricSpaceIsTopologicalSemiringContinuousStarRingStarRingAlgebraAlgebraAlgebraIsScalarTowerStarModuleContinuousSMulTopologicalSpaceContinuousFunctionalCalculusContinuousFunctionalCalculusContinuousMap.UniqueHom