Theorems · Theorem · functional analysis

QuasispectrumRestricts.cfc

∀ {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 : Field S]
  [inst_6 : StarRing S] [inst_7 : MetricSpace S] [inst_8 : IsTopologicalRing S] [inst_9 : ContinuousStar S]
  [inst_10 : NonUnitalRing A] [inst_11 : StarRing A] [inst_12 : Module S A] [inst_13 : IsScalarTower S A A]
  [inst_14 : SMulCommClass S A A] [inst_15 : Algebra R S] [inst_16 : Module R A] [IsScalarTower R S A] [StarModule R S]
  [ContinuousSMul R S] [inst_20 : TopologicalSpace A] [NonUnitalContinuousFunctionalCalculus S A q]
  [inst_22 : IsScalarTower R A A] [inst_23 : SMulCommClass R A A] (f : C(S, R)),
  Topology.IsClosedEmbedding ⇑(algebraMap R S) →
    p 0 → (∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f) → NonUnitalContinuousFunctionalCalculus R A p

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

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict

Cited by

2 results in Mathlib

Foundations

Depth 104 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemifieldStarRingMetricSpaceIsTopologicalSemiringContinuousStarFieldStarRingMetricSpaceIsTopologicalRingContinuousStarNonUnitalRingStarRingModuleIsScalarTowerSMulCommClassAlgebraModuleIsScalarTowerStarModuleContinuousSMulTopologicalSpaceNonUnitalContinuousFunctionalCalculusIsScalarTowerSMulCommClass