Theorems · Definition · functional analysis

LinearIsometryEquiv.conjStarAlgEquiv

{𝕜 : Type u_1} →
  [inst : RCLike 𝕜] →
    {H : Type u_5} →
      [inst_1 : NormedAddCommGroup H] →
        [inst_2 : InnerProductSpace 𝕜 H] →
          [inst_3 : CompleteSpace H] →
            {K : Type u_6} →
              [inst_4 : NormedAddCommGroup K] →
                [inst_5 : InnerProductSpace 𝕜 K] →
                  [inst_6 : CompleteSpace K] → (H ≃ₗᵢ[𝕜] K) → (H →L[𝕜] H) ≃⋆ₐ[𝕜] K →L[𝕜] K

An isometric linear equivalence of two Hilbert spaces induces an equivalence of ⋆-algebras of their endomorphisms. When H = K, this is exactly Unitary.conjStarAlgAut (see Unitary.conjStarAlgEquiv_unitaryLinearIsometryEquiv and Unitary.conjStarAlgAut_symm_unitaryLinearIsometryEquiv).

Defined in

Mathlib.Analysis.InnerProductSpace.Adjoint

Cited by

10 results in Mathlib

Foundations

Depth 198 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceCompleteSpaceNormedAddCommGroupInnerProductSpaceCompleteSpace