| import Mathlib |
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| namespace OAI |
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| noncomputable section |
| open scoped ComplexOrder InnerProductSpace |
| open Complex ContinuousLinearMap UniformSpace Completion |
| open scoped InnerProductSpace |
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| namespace Kadison |
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| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E] |
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| abbrev centeredSpace (Ω : E) : Submodule ℂ E := (ℂ ∙ Ω)ᗮ |
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| def center (Ω : E) : E →L[ℂ] centeredSpace Ω := |
| (centeredSpace Ω).orthogonalProjectionOnto |
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| lemma norm_center_le (Ω x : E) : ‖center Ω x‖ ≤ ‖x‖ := |
| (centeredSpace Ω).norm_orthogonalProjectionOnto_apply_le x |
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| lemma coe_center (Ω : E) (hΩ : ‖Ω‖ = 1) (x : E) : |
| (center Ω x : E) = x - inner ℂ Ω x • Ω := by |
| change (ℂ ∙ Ω)ᗮ.starProjection x = _ |
| rw [Submodule.starProjection_orthogonal_val, |
| Submodule.starProjection_unit_singleton ℂ hΩ] |
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| lemma center_vacuum (Ω : E) : center Ω Ω = 0 := by |
| apply Subtype.ext |
| change (ℂ ∙ Ω)ᗮ.starProjection Ω = (0 : E) |
| exact Submodule.starProjection_orthogonalComplement_singleton_eq_zero Ω |
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| lemma center_centered (Ω : E) (x : centeredSpace Ω) : center Ω (x : E) = x := |
| (centeredSpace Ω).orthogonalProjectionOnto_mem_subspace_eq_self x |
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| lemma denseRange_center {M : Type*} [TopologicalSpace M] (Ω : E) (v : M → E) |
| (hv : DenseRange v) : DenseRange (fun a => center Ω (v a)) := by |
| have hc : Function.Surjective (center Ω) := fun x => ⟨x, center_centered Ω x⟩ |
| exact hc.denseRange.comp hv (center Ω).continuous |
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| lemma center_isometry (Ω : E) (hΩ : ‖Ω‖ = 1) (U : E →ₗᵢ[ℂ] E) (hU : U Ω = Ω) |
| (x : E) : U (center Ω x : E) = (center Ω (U x) : E) := by |
| rw [coe_center Ω hΩ, coe_center Ω hΩ, map_sub, map_smul, hU] |
| congr 2 |
| have h := U.inner_map_map Ω x |
| rw [hU] at h |
| exact h.symm |
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| def centeredIsometry (Ω : E) (U : E →ₗᵢ[ℂ] E) (hU : U Ω = Ω) : |
| centeredSpace Ω →ₗᵢ[ℂ] centeredSpace Ω := |
| { U.toLinearMap.domRestrict (centeredSpace Ω) |>.codRestrict (centeredSpace Ω) |
| (fun x => by |
| change U (x : E) ∈ (ℂ ∙ Ω)ᗮ |
| rw [Submodule.mem_orthogonal_singleton_iff_inner_right] |
| have h := U.inner_map_map Ω (x : E) |
| rw [hU] at h |
| rw [h] |
| exact Submodule.mem_orthogonal_singleton_iff_inner_right.mp x.property) with |
| norm_map' := fun x => U.norm_map x } |
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| @[simp] lemma centeredIsometry_apply (Ω : E) (U : E →ₗᵢ[ℂ] E) (hU : U Ω = Ω) |
| (x : centeredSpace Ω) : (centeredIsometry Ω U hU x : E) = U (x : E) := rfl |
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| lemma centeredIsometry_center (Ω : E) (hΩ : ‖Ω‖ = 1) |
| (U : E →ₗᵢ[ℂ] E) (hU : U Ω = Ω) (x : E) : |
| centeredIsometry Ω U hU (center Ω x) = center Ω (U x) := by |
| apply Subtype.ext |
| exact center_isometry Ω hΩ U hU x |
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| end Kadison |
| end |
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| end OAI |
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