Theorems · Theorem · functional analysis

NonUnitalContinuousFunctionalCalculus.exists_cfc_of_predicate

∀ {R : Type u_1} {A : Type u_2} {p : outParam (A → Prop)} {inst : CommSemiring R} {inst_1 : Nontrivial R}
  {inst_2 : StarRing R} {inst_3 : MetricSpace R} {inst_4 : IsTopologicalSemiring R} {inst_5 : ContinuousStar R}
  {inst_6 : NonUnitalRing A} {inst_7 : StarRing A} {inst_8 : TopologicalSpace A} {inst_9 : Module R A}
  {inst_10 : IsScalarTower R A A} {inst_11 : SMulCommClass R A A} [self : NonUnitalContinuousFunctionalCalculus R A p]
  (a : A),
  p a →
    ∃ φ,
      Continuous ⇑φ ∧
        Function.Injective ⇑φ ∧
          φ { toContinuousMap := ContinuousMap.restrict (quasispectrum R a) (ContinuousMap.id R), map_zero' := ⋯ } = a ∧
            (∀ (f : ContinuousMapZero (↑(quasispectrum R a)) R), quasispectrum R (φ f) = Set.range ⇑f) ∧
              ∀ (f : ContinuousMapZero (↑(quasispectrum R a)) R), p (φ f)

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Cited by

5 results in Mathlib

Foundations

Depth 98 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NonUnitalContinuousFunctionalCalculus