Theorems · Theorem · functional analysis

StarAlgHom.map_cfc

∀ {R : Type u_2} {S : Type u_3} {A : Type u_4} {B : Type u_5} {p : A → Prop} {q : B → Prop} [inst : CommSemiring R]
  [inst_1 : StarRing R] [inst_2 : MetricSpace R] [inst_3 : IsTopologicalSemiring R] [inst_4 : ContinuousStar R]
  [inst_5 : Ring A] [inst_6 : StarRing A] [inst_7 : TopologicalSpace A] [inst_8 : Algebra R A] [inst_9 : Ring B]
  [inst_10 : StarRing B] [inst_11 : TopologicalSpace B] [inst_12 : Algebra R B] [inst_13 : CommSemiring S]
  [inst_14 : Algebra R S] [inst_15 : Algebra S A] [inst_16 : Algebra S B] [IsScalarTower R S A] [IsScalarTower R S B]
  [inst_19 : ContinuousFunctionalCalculus R A p] [inst_20 : ContinuousFunctionalCalculus R B q]
  [ContinuousMap.UniqueHom R B] (φ : A →⋆ₐ[S] B) (f : R → R) (a : A),
  autoParam (ContinuousOn f (spectrum R a)) StarAlgHom.map_cfc._auto_1 →
    autoParam (Continuous ⇑φ) StarAlgHom.map_cfc._auto_3 →
      autoParam (p a) StarAlgHom.map_cfc._auto_5 →
        autoParam (q (φ a)) StarAlgHom.map_cfc._auto_7 → φ (cfc f a) = cfc f (φ a)

Star algebra homomorphisms commute with the continuous functional calculus. This version is specialized to A →⋆ₐ[S] B to allow for dot notation.

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Cited by

2 results in Mathlib

Foundations

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

Assumes

CommSemiringStarRingMetricSpaceIsTopologicalSemiringContinuousStarRingStarRingTopologicalSpaceAlgebraRingStarRingTopologicalSpaceAlgebraCommSemiringAlgebraAlgebraAlgebraIsScalarTowerIsScalarTowerContinuousFunctionalCalculusContinuousFunctionalCalculusContinuousMap.UniqueHom