Theorems · Definition · functional analysis

WeakDual.CharacterSpace.compContinuousMap

{A : Type u_1} →
  {B : Type u_2} →
    {𝕜 : Type u_4} →
      [inst : NontriviallyNormedField 𝕜] →
        [inst_1 : NormedRing A] →
          [inst_2 : NormedAlgebra 𝕜 A] →
            [CompleteSpace A] →
              [inst_4 : StarRing A] →
                [inst_5 : NormedRing B] →
                  [inst_6 : NormedAlgebra 𝕜 B] →
                    [CompleteSpace B] →
                      [inst_8 : StarRing B] →
                        (A →⋆ₐ[𝕜] B) → C(↑(WeakDual.characterSpace 𝕜 B), ↑(WeakDual.characterSpace 𝕜 A))

The functorial map taking ψ : A →⋆ₐ[ℂ] B to a continuous function characterSpace ℂ B → characterSpace ℂ A obtained by pre-composition with ψ.

Defined in

Mathlib.Analysis.CStarAlgebra.GelfandDuality

Cited by

5 results in Mathlib

Foundations

Depth 181 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedRingNormedAlgebraCompleteSpaceStarRingNormedRingNormedAlgebraCompleteSpaceStarRing