lean/OAI/Analysis/FreeGroupFactor/NormalExtension.lean

import OAI.Analysis.FreeGroupFactor.MomentFlow

namespace OAI

namespace FreeGroupFactorMain

noncomputable section

universe uG uH

open Set Metric Filter
open scoped NNReal Topology BigOperators ContDiff

open scoped ComplexConjugate

abbrev deltaOne {G : Type uG} [Group G] : GroupHilbert G := by
  classical
  exact lp.single 2 1 1

theorem deltaOne_eq {G : Type uG} [Group G] [DecidableEq G] :
    (deltaOne : GroupHilbert G) = lp.single 2 1 1 := by
  ext x
  simp [deltaOne, lp.single_apply, Pi.single_apply]

theorem groupVN_matrix {G : Type uG} [Group G] [DecidableEq G]
    (T : groupVonNeumann G) (x y : G) :
    ((T : GroupOperator G) (lp.single 2 y 1)) x =
      ((T : GroupOperator G) deltaOne) (x * y⁻¹) := by
  have hr : rightRegular y⁻¹ deltaOne = lp.single 2 y 1 := by
    ext b
    simp [deltaOne, lp.single_apply, Pi.single_apply, mul_inv_eq_one]
  rw [← hr]
  have hc := congrArg (fun f : GroupOperator G => f deltaOne)
    (groupVN_commute_right T y⁻¹)
  simpa only [mul_apply_eq_comp, rightRegular_apply] using
    congrArg (fun v : GroupHilbert G => v x) hc.symm

theorem groupVN_star_coefficient {G : Type uG} [Group G] [DecidableEq G]
    (T : groupVonNeumann G) (x : G) :
    ((star T : GroupOperator G) deltaOne) x =
      conj (((T : GroupOperator G) deltaOne) x⁻¹) := by
  calc
    ((star T : GroupOperator G) deltaOne) x =
        inner ℂ (lp.single 2 x (1 : ℂ)) ((star T : GroupOperator G) deltaOne) := by
      simp only [lp.inner_single_left, RCLike.inner_apply, map_one, mul_one]
    _ = inner ℂ ((T : GroupOperator G) (lp.single 2 x 1)) deltaOne := by
      exact ContinuousLinearMap.adjoint_inner_right _ _ _
    _ = conj (((T : GroupOperator G) (lp.single 2 x 1)) 1) := by
      simp only [deltaOne, lp.inner_single_right, RCLike.inner_apply, one_mul]
    _ = _ := by rw [groupVN_matrix]; simp

theorem canonicalTrace_star_mul_inner {G : Type uG} [Group G] [DecidableEq G]
    (T U : groupVonNeumann G) :
    canonicalTrace (star T * U) = inner ℂ
      ((T : GroupOperator G) deltaOne) ((U : GroupOperator G) deltaOne) := by
  classical
  rw [canonicalTrace_eq_inner, ← deltaOne_eq]
  change inner ℂ deltaOne (star (T : GroupOperator G) ((U : GroupOperator G) deltaOne)) = _
  rw [ContinuousLinearMap.star_eq_adjoint, ContinuousLinearMap.adjoint_inner_right]

theorem canonicalTrace_mul {G : Type uG} [Group G] [DecidableEq G]
    (T U : groupVonNeumann G) : canonicalTrace (T * U) = canonicalTrace (U * T) := by
  have he (T U : groupVonNeumann G) :
      canonicalTrace (T * U) = ∑' x : G,
        ((T : GroupOperator G) deltaOne) x⁻¹ * ((U : GroupOperator G) deltaOne) x := by
    rw [← star_star T, canonicalTrace_star_mul_inner]
    change (∑' x, inner ℂ (((star T : GroupOperator G) deltaOne) x)
      (((U : GroupOperator G) deltaOne) x)) = _
    apply tsum_congr
    intro x
    rw [groupVN_star_coefficient]
    simp [RCLike.inner_apply, mul_comm]
  rw [he, he]
  simpa only [Equiv.inv_apply, inv_inv, mul_comm] using
    ((Equiv.inv G).tsum_eq (fun x =>
      ((T : GroupOperator G) deltaOne) x⁻¹ * ((U : GroupOperator G) deltaOne) x)).symm

def hilbertSequenceMap {G : Type uG} {H : Type uH} (e : GroupHilbert G ≃ₗᵢ[ℂ] GroupHilbert H)
    (ξ : lp (fun _ : ℕ => GroupHilbert G) 2) :
    lp (fun _ : ℕ => GroupHilbert H) 2 :=
  ⟨fun n => e (ξ n), Memℓp.of_norm (by simpa only [e.norm_map] using (lp.memℓp ξ).norm)⟩

theorem spatial_ultraweak_continuous {G : Type uG} {H : Type uH}
    (e : GroupHilbert G ≃ₗᵢ[ℂ] GroupHilbert H) :
    @Continuous _ _ (ultraweakOperatorTopology G) (ultraweakOperatorTopology H)
      e.conjStarAlgEquiv := by
  apply continuous_iInf_rng.mpr
  intro ξ
  apply continuous_iInf_rng.mpr
  intro η
  apply continuous_induced_rng.mpr
  have heq :
      (fun T : GroupOperator G => ∑' k : ℕ, inner ℂ (η k) (e.conjStarAlgEquiv T (ξ k))) =
      (fun T : GroupOperator G => ∑' k : ℕ,
        inner ℂ (hilbertSequenceMap e.symm η k) (T (hilbertSequenceMap e.symm ξ k))) := by
    funext T
    apply tsum_congr
    intro k
    change inner ℂ (η k) (e (T (e.symm (ξ k)))) =
      inner ℂ (e.symm (η k)) (T (e.symm (ξ k)))
    rw [← e.inner_map_map]
    simp
  change @Continuous _ _ (ultraweakOperatorTopology G) _
    (fun T : GroupOperator G => ∑' k : ℕ, inner ℂ (η k) (e.conjStarAlgEquiv T (ξ k)))
  rw [heq]
  exact continuous_iInf_dom (i := hilbertSequenceMap e.symm ξ)
    (continuous_iInf_dom (i := hilbertSequenceMap e.symm η) continuous_induced_dom)

def spatialGroupIso {G : Type uG} {H : Type uH} [Group G] [Group H]
    (e : GroupHilbert G ≃ₗᵢ[ℂ] GroupHilbert H)
    (hVN : ∀ T : GroupOperator G,
      T ∈ groupVonNeumann G ↔ e.conjStarAlgEquiv T ∈ groupVonNeumann H) :
    groupVonNeumann G ≃⋆ₐ[ℂ] groupVonNeumann H where
  toFun T := ⟨e.conjStarAlgEquiv T, (hVN T).mp T.property⟩
  invFun T := ⟨e.conjStarAlgEquiv.symm T, (hVN _).mpr (by rw [e.conjStarAlgEquiv.apply_symm_apply]; exact T.property)⟩
  left_inv T := Subtype.ext (e.conjStarAlgEquiv.symm_apply_apply T)
  right_inv T := Subtype.ext (e.conjStarAlgEquiv.apply_symm_apply T)
  map_mul' T U := Subtype.ext (map_mul e.conjStarAlgEquiv (T : GroupOperator G) (U : GroupOperator G))
  map_add' T U := Subtype.ext (map_add e.conjStarAlgEquiv (T : GroupOperator G) (U : GroupOperator G))
  map_star' T := Subtype.ext (map_star e.conjStarAlgEquiv (T : GroupOperator G))
  map_smul' c T := Subtype.ext (map_smul e.conjStarAlgEquiv c (T : GroupOperator G))

theorem spatialGroupIso_normal {G : Type uG} {H : Type uH} [Group G] [Group H]
    (e : GroupHilbert G ≃ₗᵢ[ℂ] GroupHilbert H)
    (hVN : ∀ T : GroupOperator G,
      T ∈ groupVonNeumann G ↔ e.conjStarAlgEquiv T ∈ groupVonNeumann H) :
    @Continuous _ _ (groupUltraweakTopology G) (groupUltraweakTopology H)
      (spatialGroupIso e hVN) := by
  let : TopologicalSpace (GroupOperator G) := ultraweakOperatorTopology G
  let : TopologicalSpace (GroupOperator H) := ultraweakOperatorTopology H
  let : TopologicalSpace (groupVonNeumann G) := groupUltraweakTopology G
  let : TopologicalSpace (groupVonNeumann H) := groupUltraweakTopology H
  apply continuous_induced_rng.mpr
  have hinc : Continuous (fun T : groupVonNeumann G => (T : GroupOperator G)) :=
    continuous_induced_dom
  exact (spatial_ultraweak_continuous e).comp hinc

theorem normalTracialIso_of_spatial {G : Type uG} {H : Type uH} [Group G] [Group H]
    (e : GroupHilbert G ≃ₗᵢ[ℂ] GroupHilbert H)
    (hδ : e deltaOne = deltaOne)
    (hVN : ∀ T : GroupOperator G,
      T ∈ groupVonNeumann G ↔ e.conjStarAlgEquiv T ∈ groupVonNeumann H) :
    HasNormalTracialIso G H := by
  classical
  refine ⟨spatialGroupIso e hVN, fun T => ?_, spatialGroupIso_normal e hVN⟩
  rw [canonicalTrace_eq_inner, canonicalTrace_eq_inner]
  change inner ℂ deltaOne (e ((T : GroupOperator G) (e.symm deltaOne))) =
    inner ℂ deltaOne ((T : GroupOperator G) deltaOne)
  rw [← hδ, e.symm_apply_apply, e.inner_map_map]

open scoped ComplexConjugate ENNReal

 theorem operator_matrix_ext {G : Type uG} [DecidableEq G] (T U : GroupOperator G)
    (h : ∀ g, T (lp.single 2 g 1) = U (lp.single 2 g 1)) : T = U := by
  classical
  apply ContinuousLinearMap.ext
  intro ξ
  have hT := (lp.hasSum_single (by norm_num : (2 : ℝ≥0∞) ≠ ⊤) ξ).mapL T
  have hU := (lp.hasSum_single (by norm_num : (2 : ℝ≥0∞) ≠ ⊤) ξ).mapL U
  have he (g : G) : T (lp.single 2 g (ξ g)) = U (lp.single 2 g (ξ g)) := by
    have hc : lp.single (E := fun _ : G => ℂ) 2 g (ξ g) =
        (ξ g) • lp.single (E := fun _ : G => ℂ) 2 g 1 := by
      ext b
      simp [lp.single_apply, Pi.single_apply]
    rw [hc, map_smul, map_smul, h]
  simp only [he] at hT
  exact hT.unique hU

 theorem operator_coeff_tsum {G : Type uG} [DecidableEq G] (T : GroupOperator G)
    (ξ : GroupHilbert G) (x : G) :
    (T ξ) x = ∑' y : G, (T (lp.single 2 y 1)) x * ξ y := by
  calc
    (T ξ) x = inner ℂ (lp.single 2 x 1) (T ξ) := by simp [lp.inner_single_left]
    _ = inner ℂ (star T (lp.single 2 x 1)) ξ := by
      rw [ContinuousLinearMap.star_eq_adjoint, ContinuousLinearMap.adjoint_inner_left]
    _ = ∑' y : G, (T (lp.single 2 y 1)) x * ξ y := by
      rw [lp.inner_eq_tsum]
      apply tsum_congr
      intro y
      have he : conj ((star T (lp.single 2 x 1)) y) = (T (lp.single 2 y 1)) x := by
        calc
          _ = inner ℂ (star T (lp.single 2 x 1)) (lp.single 2 y 1) := by
            simp [lp.inner_single_right]
          _ = inner ℂ (lp.single 2 x 1) (T (lp.single 2 y 1)) := by
            rw [ContinuousLinearMap.star_eq_adjoint, ContinuousLinearMap.adjoint_inner_left]
          _ = _ := by simp [lp.inner_single_left]
      simp only [RCLike.inner_apply, he, mul_comm]

 theorem matrix_of_commute_right {G : Type uG} [Group G] [DecidableEq G]
    (T : GroupOperator G) (hT : ∀ g, rightRegular g * T = T * rightRegular g)
    (x y : G) :
    (T (lp.single 2 y 1)) x = (T deltaOne) (x * y⁻¹) := by
  have hr : rightRegular y⁻¹ deltaOne = lp.single 2 y 1 := by
    ext b
    simp [deltaOne, lp.single_apply, Pi.single_apply, mul_inv_eq_one]
  rw [← hr]
  have hc := congrArg (fun f : GroupOperator G => f deltaOne) (hT y⁻¹)
  simpa only [mul_apply_eq_comp, rightRegular_apply] using
    congrArg (fun v : GroupHilbert G => v x) hc.symm

 theorem matrix_of_commute_left {G : Type uG} [Group G] [DecidableEq G]
    (T : GroupOperator G) (hT : ∀ g, leftRegular g * T = T * leftRegular g)
    (x y : G) :
    (T (lp.single 2 y 1)) x = (T deltaOne) (y⁻¹ * x) := by
  have hl : leftRegular y deltaOne = lp.single 2 y 1 := by
    ext b
    simp [deltaOne, lp.single_apply, Pi.single_apply, inv_mul_eq_one, eq_comm]
  rw [← hl]
  have hc := congrArg (fun f : GroupOperator G => f deltaOne) (hT y)
  simpa only [mul_apply_eq_comp, leftRegular_apply] using
    congrArg (fun v : GroupHilbert G => v x) hc.symm

 theorem mem_groupVN_iff_commute_right {G : Type uG} [Group G]
    (T : GroupOperator G) : T ∈ groupVonNeumann G ↔
      ∀ g, rightRegular g * T = T * rightRegular g := by
  classical
  refine ⟨fun h g => groupVN_commute_right ⟨T,h⟩ g, ?_⟩
  intro hT
  have hc (U : GroupOperator G) (hU : ∀ g, leftRegular g * U = U * leftRegular g) :
      U * T = T * U := by
    apply operator_matrix_ext
    intro k
    ext x
    have hleft : ((U * T) (lp.single 2 k 1)) x =
        ∑' y : G, (U deltaOne) (y⁻¹ * x) * (T deltaOne) (y * k⁻¹) := by
      change (U (T (lp.single 2 k 1))) x = _
      rw [operator_coeff_tsum U]
      apply tsum_congr
      intro y
      rw [matrix_of_commute_left U hU x y, matrix_of_commute_right T hT y k]
    have hright : ((T * U) (lp.single 2 k 1)) x =
        ∑' y : G, (T deltaOne) (x * y⁻¹) * (U deltaOne) (k⁻¹ * y) := by
      change (T (U (lp.single 2 k 1))) x = _
      rw [operator_coeff_tsum T]
      apply tsum_congr
      intro y
      rw [matrix_of_commute_right T hT x y, matrix_of_commute_left U hU y k]
    rw [hleft, hright]
    let e : G ≃ G := ((Equiv.inv G).trans (Equiv.mulLeft x)).trans (Equiv.mulRight k)
    have hs := e.tsum_eq (fun y => (U deltaOne) (y⁻¹ * x) * (T deltaOne) (y * k⁻¹))
    change (∑' c : G, (U deltaOne) ((x * c⁻¹ * k)⁻¹ * x) *
      (T deltaOne) (x * c⁻¹ * k * k⁻¹)) = _ at hs
    simpa only [mul_inv_rev, inv_inv, mul_assoc, inv_mul_cancel,
      mul_inv_cancel, mul_one, mul_comm] using hs.symm
  rw [groupVonNeumann, StarSubalgebra.mem_centralizer_iff]
  intro U hU
  have hu : ∀ g, leftRegular g * U = U * leftRegular g := by
    intro g
    exact ((StarSubalgebra.mem_centralizer_iff ℂ).mp hU _
      (leftRegular_mem_leftTranslations g)).1
  refine ⟨hc U hu, hc (star U) ?_⟩
  intro g
  have hh := congrArg star (hu g⁻¹)
  simpa only [star_mul, leftRegular_star, inv_inv] using hh.symm

open scoped ComplexConjugate

 theorem rightRegular_single {G : Type uG} [Group G] [DecidableEq G] (g y : G) :
    rightRegular g (lp.single 2 y (1 : ℂ)) = lp.single 2 (y * g⁻¹) 1 := by
  ext x
  simp only [rightRegular_apply, lp.single_apply, Pi.single_apply]
  congr 1
  exact propext (eq_mul_inv_iff_mul_eq.symm)

 theorem mem_groupVN_of_matrix_invariant {G : Type uG} [Group G] [DecidableEq G]
    (T : GroupOperator G)
    (hT : ∀ x y g : G, (T (lp.single 2 (y*g) 1)) (x*g) =
      (T (lp.single 2 y 1)) x) : T ∈ groupVonNeumann G := by
  rw [mem_groupVN_iff_commute_right]
  intro g
  apply operator_matrix_ext
  intro y
  ext x
  simp only [mul_apply_eq_comp, rightRegular_apply]
  change (T (lp.single 2 y 1)) (x*g) = (T (rightRegular g (lp.single 2 y 1))) x
  rw [rightRegular_single]
  simpa only [mul_inv_cancel_right] using (hT (x*g) y g⁻¹).symm

 theorem haar_orbit_orthonormal {G : Type uG} {H : Type uH} [Group G] [Group H] [DecidableEq G]
    (ρ : G →* groupVonNeumann H)
    (hstar : ∀ g, star (ρ g) = ρ g⁻¹)
    (htrace : ∀ g, canonicalTrace (ρ g) = if g = 1 then 1 else 0) :
    Orthonormal ℂ (fun g => ((ρ g : groupVonNeumann H) : GroupOperator H) deltaOne) := by
  classical
  rw [orthonormal_iff_ite]
  intro x y
  rw [← canonicalTrace_star_mul_inner, hstar, ← map_mul, htrace]
  simp only [inv_mul_eq_one]

 theorem spatial_of_haar_total {G : Type uG} {H : Type uH} [Group G] [Group H] [DecidableEq G]
    (ρ : G →* groupVonNeumann H)
    (hstar : ∀ g, star (ρ g) = ρ g⁻¹)
    (htrace : ∀ g, canonicalTrace (ρ g) = if g = 1 then 1 else 0)
    (htotal : ⊤ ≤ (Submodule.span ℂ (Set.range
      (fun g => ((ρ g : groupVonNeumann H) : GroupOperator H) deltaOne))).topologicalClosure) :
    ∃ e : GroupHilbert G ≃ₗᵢ[ℂ] GroupHilbert H,
      e deltaOne = deltaOne ∧
      (∀ g, e.conjStarAlgEquiv (leftRegular g) = (ρ g : GroupOperator H)) ∧
      ∀ T : GroupOperator G, T ∈ groupVonNeumann G ↔
        e.conjStarAlgEquiv T ∈ groupVonNeumann H := by
  classical
  let v : G → GroupHilbert H := fun g => ((ρ g : groupVonNeumann H) : GroupOperator H) deltaOne
  have hon : Orthonormal ℂ v := haar_orbit_orthonormal ρ hstar htrace
  let e : GroupHilbert G ≃ₗᵢ[ℂ] GroupHilbert H :=
    (hon.isHilbertSum htotal).linearIsometryEquiv.symm
  have he (g : G) : e (lp.single 2 g 1) = v g :=
    hon.linearIsometryEquiv_symm_apply_single_one htotal g
  have hδ : e deltaOne = deltaOne := by
    rw [deltaOne_eq, he]
    simp [v]
  have hleft (g : G) : e.conjStarAlgEquiv (leftRegular g) = (ρ g : GroupOperator H) := by
    apply e.conjStarAlgEquiv.symm.injective
    change e.conjStarAlgEquiv.symm (e.conjStarAlgEquiv (leftRegular g)) =
      e.conjStarAlgEquiv.symm (ρ g : GroupOperator H)
    rw [e.conjStarAlgEquiv.symm_apply_apply]
    apply operator_matrix_ext
    intro y
    apply e.injective
    change e (leftRegular g (lp.single 2 y 1)) =
      e (e.symm ((ρ g : GroupOperator H) (e (lp.single 2 y 1))))
    rw [e.apply_symm_apply, he]
    have hl : leftRegular g (lp.single 2 y (1 : ℂ)) = lp.single 2 (g*y) 1 := by
      ext x
      simp only [leftRegular_apply, lp.single_apply, Pi.single_apply]
      congr 1
      exact propext (inv_mul_eq_iff_eq_mul)
    rw [hl, he]
    change ((ρ (g*y) : groupVonNeumann H) : GroupOperator H) deltaOne = _
    rw [map_mul]
    rfl
  refine ⟨e, hδ, hleft, ?_⟩
  intro T
  constructor
  · intro hT
    rw [mem_groupVN_iff_commute_right]
    intro g
    apply e.conjStarAlgEquiv.symm.injective
    change e.conjStarAlgEquiv.symm (rightRegular g * e.conjStarAlgEquiv T) =
      e.conjStarAlgEquiv.symm (e.conjStarAlgEquiv T * rightRegular g)
    rw [map_mul, map_mul, e.conjStarAlgEquiv.symm_apply_apply]
    have hc : e.conjStarAlgEquiv.symm (rightRegular g) ∈
        StarSubalgebra.centralizer ℂ (leftTranslations G) := by
      rw [StarSubalgebra.mem_centralizer_iff, leftTranslations_eq_range]
      rintro _ ⟨x,rfl⟩
      constructor
      · apply e.conjStarAlgEquiv.injective
        change e.conjStarAlgEquiv (leftRegular _ * e.conjStarAlgEquiv.symm (rightRegular g)) =
          e.conjStarAlgEquiv (e.conjStarAlgEquiv.symm (rightRegular g) * leftRegular _)
        rw [map_mul, map_mul, e.conjStarAlgEquiv.apply_symm_apply, hleft]
        exact (groupVN_commute_right (ρ x) g).symm
      · rw [leftRegular_star]
        apply e.conjStarAlgEquiv.injective
        change e.conjStarAlgEquiv (leftRegular _ * e.conjStarAlgEquiv.symm (rightRegular g)) =
          e.conjStarAlgEquiv (e.conjStarAlgEquiv.symm (rightRegular g) * leftRegular _)
        rw [map_mul, map_mul, e.conjStarAlgEquiv.apply_symm_apply, hleft]
        exact (groupVN_commute_right (ρ x⁻¹) g).symm
    exact ((StarSubalgebra.mem_centralizer_iff ℂ).mp hT _ hc).1
  · intro hT
    let U : groupVonNeumann H := ⟨e.conjStarAlgEquiv T, hT⟩
    have hm (x y : G) : (T (lp.single 2 y 1)) x =
        canonicalTrace (star (ρ x) * U * ρ y) := by
      calc
        (T (lp.single 2 y 1)) x = inner ℂ (lp.single 2 x 1) (T (lp.single 2 y 1)) := by
          simp [lp.inner_single_left]
        _ = inner ℂ (v x) ((U : GroupOperator H) (v y)) := by
          rw [← he x, ← he y, ← e.inner_map_map]
          change inner ℂ (e (lp.single 2 x 1)) (e (T (lp.single 2 y 1))) =
            inner ℂ (e (lp.single 2 x 1)) (e (T (e.symm (e (lp.single 2 y 1)))))
          rw [e.symm_apply_apply]
        _ = canonicalTrace (star (ρ x) * U * ρ y) := by
          rw [mul_assoc, canonicalTrace_star_mul_inner]
          rfl
    apply mem_groupVN_of_matrix_invariant T
    intro x y g
    rw [hm, hm, map_mul, map_mul, star_mul]
    have hc := canonicalTrace_mul (star (ρ g) * star (ρ x) * U * ρ y) (ρ g)
    simp only [← mul_assoc]
    rw [hc]
    simp only [hstar, ← mul_assoc, ← map_mul, mul_inv_cancel, one_mul]

 theorem normalTracialIso_of_haar_total {G : Type uG} {H : Type uH} [Group G] [Group H] [DecidableEq G]
    (ρ : G →* groupVonNeumann H)
    (hstar : ∀ g, star (ρ g) = ρ g⁻¹)
    (htrace : ∀ g, canonicalTrace (ρ g) = if g = 1 then 1 else 0)
    (htotal : ⊤ ≤ (Submodule.span ℂ (Set.range
      (fun g => ((ρ g : groupVonNeumann H) : GroupOperator H) deltaOne))).topologicalClosure) :
    HasNormalTracialIso G H := by
  obtain ⟨e,hδ,_hρ,hVN⟩ := spatial_of_haar_total ρ hstar htrace htotal
  exact normalTracialIso_of_spatial e hδ hVN

 theorem iso_extension_of_haar_total {G : Type uG} {H : Type uH} [Group G] [Group H] [DecidableEq G]
    (ρ : G →* groupVonNeumann H)
    (hstar : ∀ g, star (ρ g) = ρ g⁻¹)
    (htrace : ∀ g, canonicalTrace (ρ g) = if g = 1 then 1 else 0)
    (htotal : ⊤ ≤ (Submodule.span ℂ (Set.range
      (fun g => ((ρ g : groupVonNeumann H) : GroupOperator H) deltaOne))).topologicalClosure) :
    ∃ Φ : groupVonNeumann G ≃⋆ₐ[ℂ] groupVonNeumann H,
      (∀ T, canonicalTrace (Φ T) = canonicalTrace T) ∧
      @Continuous _ _ (groupUltraweakTopology G) (groupUltraweakTopology H) Φ ∧
      ∀ g, Φ (groupUnitary g) = ρ g := by
  classical
  obtain ⟨e,hδ,hρ,hVN⟩ := spatial_of_haar_total ρ hstar htrace htotal
  refine ⟨spatialGroupIso e hVN, ?_, spatialGroupIso_normal e hVN, ?_⟩
  · intro T
    rw [canonicalTrace_eq_inner, canonicalTrace_eq_inner, ← deltaOne_eq, ← deltaOne_eq]
    change inner ℂ deltaOne (e ((T : GroupOperator G) (e.symm deltaOne))) =
      inner ℂ deltaOne ((T : GroupOperator G) deltaOne)
    rw [← hδ, e.symm_apply_apply, e.inner_map_map]
  · intro g
    apply Subtype.ext
    exact hρ g

end

end FreeGroupFactorMain

end OAI