Theorems · Theorem · functional analysis

SpectrumRestricts.starAlgHom_injective

∀ {R : Type u_1} {S : Type u_2} {A : Type u_3} [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] {a : A}
  {φ : C(↑(spectrum S a), S) →⋆ₐ[S] A},
  Function.Injective ⇑φ →
    ∀ {f : C(S, R)} (h : SpectrumRestricts a ⇑f),
      Function.Injective ⇑(algebraMap R S) → Function.Injective ⇑(SpectrumRestricts.starAlgHom φ h)

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict

Cited by

1 results in Mathlib

Foundations

Depth 96 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemifieldStarRingMetricSpaceIsTopologicalSemiringContinuousStarSemifieldStarRingMetricSpaceIsTopologicalSemiringContinuousStarRingStarRingAlgebraAlgebraAlgebraIsScalarTowerStarModuleContinuousSMul