Theorems · Theorem · functional analysis

QuasispectrumRestricts.nonUnitalStarAlgHom_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 : 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] [inst_17 : IsScalarTower R S A] [inst_18 : StarModule R S]
  [inst_19 : ContinuousSMul R S] {a : A} {φ : ContinuousMapZero (↑(quasispectrum S a)) S →⋆ₙₐ[S] A},
  Function.Injective ⇑φ →
    ∀ {f : C(S, R)} (h : QuasispectrumRestricts a ⇑f),
      Function.Injective ⇑(algebraMap R S) → Function.Injective ⇑(QuasispectrumRestricts.nonUnitalStarAlgHom φ h)

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict

Cited by

1 results in Mathlib

Foundations

Depth 102 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemifieldStarRingMetricSpaceIsTopologicalSemiringContinuousStarFieldStarRingMetricSpaceIsTopologicalRingContinuousStarNonUnitalRingStarRingModuleIsScalarTowerSMulCommClassAlgebraModuleIsScalarTowerStarModuleContinuousSMul