Theorems · Theorem · functional analysis

SpectrumRestricts.closedEmbeddingCFC

∀ {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]
  [ClosedEmbeddingContinuousFunctionalCalculus S A q] [ContinuousMap.UniqueHom R A] [CompleteSpace R] (f : C(S, R)),
  IsUniformEmbedding ⇑(algebraMap R S) →
    p 0 → (∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f) → ClosedEmbeddingContinuousFunctionalCalculus R A p

Given a ContinuousFunctionalCalculus S A q. If we form the predicate p for a : A characterized by: q a and the spectrum of a restricts to the scalar subring R via f : C(S, R), then we can get a restricted functional calculus ContinuousFunctionalCalculus R A p.

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict

Cited by

0 results in Mathlib

Foundations

Depth 153 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemifieldStarRingMetricSpaceIsTopologicalSemiringContinuousStarSemifieldStarRingMetricSpaceIsTopologicalSemiringContinuousStarRingStarRingAlgebraAlgebraAlgebraIsScalarTowerStarModuleContinuousSMulTopologicalSpaceClosedEmbeddingContinuousFunctionalCalculusContinuousMap.UniqueHomCompleteSpace