Datasets:

roseking
/

openAI-math-dataset

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

	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 (yg) 1)) (xg) =
	(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