Theorems · Theorem · functional analysis

StarAlgHom.ext_continuousMap

∀ {R : Type u_1} {A : Type u_2} [inst : CommSemiring R] [inst_1 : StarRing R] [inst_2 : MetricSpace R]
  [inst_3 : IsTopologicalSemiring R] [inst_4 : ContinuousStar R] [inst_5 : TopologicalSpace A] [inst_6 : Ring A]
  [inst_7 : StarRing A] [inst_8 : Algebra R A] [ContinuousMap.UniqueHom R A] (a : A) [CompactSpace ↑(spectrum R a)]
  (φ ψ : C(↑(spectrum R a), R) →⋆ₐ[R] A),
  Continuous ⇑φ →
    Continuous ⇑ψ →
      φ (ContinuousMap.restrict (spectrum R a) (ContinuousMap.id R)) =
          ψ (ContinuousMap.restrict (spectrum R a) (ContinuousMap.id R)) →
        φ = ψ

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Cited by

1 results in Mathlib

Foundations

Depth 94 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringStarRingMetricSpaceIsTopologicalSemiringContinuousStarTopologicalSpaceRingStarRingAlgebraContinuousMap.UniqueHomCompactSpace