Theorems · Theorem · functional analysis

SpectrumRestricts.isometric_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 : 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 : MetricSpace A]
  [IsometricContinuousFunctionalCalculus S A q] [CompleteSpace R] [ContinuousMap.UniqueHom R A] (f : C(S, R)),
  Isometry ⇑(algebraMap R S) →
    p 0 → (∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f) → IsometricContinuousFunctionalCalculus R A p

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Cited by

0 results in Mathlib

Foundations

Depth 154 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemifieldStarRingMetricSpaceIsTopologicalSemiringContinuousStarSemifieldStarRingMetricSpaceIsTopologicalSemiringContinuousStarRingStarRingAlgebraAlgebraAlgebraIsScalarTowerStarModuleContinuousSMulMetricSpaceIsometricContinuousFunctionalCalculusCompleteSpaceContinuousMap.UniqueHom