Theorems · Theorem · functional analysis

QuasispectrumRestricts.nonUnitalStarAlgHom_apply

∀ {R : Type u} {S : Type v} {A : Type w} [inst : Semifield R] [inst_1 : StarRing R] [inst_2 : TopologicalSpace R]
  [inst_3 : IsTopologicalSemiring R] [inst_4 : ContinuousStar R] [inst_5 : Field S] [inst_6 : StarRing S]
  [inst_7 : TopologicalSpace S] [inst_8 : IsTopologicalRing S] [inst_9 : ContinuousStar S] [inst_10 : NonUnitalRing A]
  [inst_11 : StarRing A] [inst_12 : Algebra R S] [inst_13 : Module R A] [inst_14 : Module S A]
  [inst_15 : IsScalarTower S A A] [inst_16 : SMulCommClass S A 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) {f : C(S, R)} (h : QuasispectrumRestricts a ⇑f)
  (a_1 : ContinuousMapZero (↑(quasispectrum R a)) R),
  (QuasispectrumRestricts.nonUnitalStarAlgHom φ h) a_1 =
    φ
      ({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯, map_zero' := ⋯ }.comp
        (a_1.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯, map_zero' := ⋯ }))

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict

Cited by

4 results in Mathlib

Foundations

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

Assumes

SemifieldStarRingTopologicalSpaceIsTopologicalSemiringContinuousStarFieldStarRingTopologicalSpaceIsTopologicalRingContinuousStarNonUnitalRingStarRingAlgebraModuleModuleIsScalarTowerSMulCommClassIsScalarTowerStarModuleContinuousSMul