Datasets:

roseking
/

openAI-math-dataset

import Mathlib.Analysis.CStarAlgebra.Hom
import Mathlib.Analysis.LocallyConvex.WeakSpace
import Mathlib.Topology.Semicontinuity.Basic
import OAI.Analysis.FactorGeneration.Bicommutant
import OAI.Analysis.FactorGeneration.CornerTrace

namespace OAI

noncomputable section

namespace Generator

/-! Multiplication and adjoint limits in L2, with the necessary uniform
operator norm bounds stated explicitly. -/
namespace Tracial
open Filter
open scoped Topology
open UltrapowerBridge
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
variable {M : VonNeumannAlgebra H} (τ : TracialRepresentation M)

theorem tendsto_vector_iff_norm₂ {I : Type*} (l : Filter I)
    (a : I → M.toStarSubalgebra) (b : M.toStarSubalgebra) :
    Tendsto (fun i => (a i : H →L[ℂ] H) τ.vector) l (𝓝 ((b : H →L[ℂ] H) τ.vector)) ↔
    Tendsto (fun i => τ.norm₂ (a i-b)) l (𝓝 0) := by
  simpa only [dist_eq_norm,TracialRepresentation.norm₂_sub] using
    (tendsto_iff_dist_tendsto_zero (f := fun i => (a i : H →L[ℂ] H) τ.vector)
      (a := (b : H →L[ℂ] H) τ.vector))

theorem norm₂_limit_unique {I : Type*} {l : Filter I} [l.NeBot]
    (a : I → M.toStarSubalgebra) {b c : M.toStarSubalgebra}
    (hb : Tendsto (fun i => τ.norm₂ (a i-b)) l (𝓝 0))
    (hc : Tendsto (fun i => τ.norm₂ (a i-c)) l (𝓝 0)) : b=c := by
  have he := tendsto_nhds_unique ((tendsto_vector_iff_norm₂ τ l a b).mpr hb)
    ((tendsto_vector_iff_norm₂ τ l a c).mpr hc)
  apply sub_eq_zero.mp
  apply τ.separating
  change (b : H →L[ℂ] H) τ.vector-(c : H →L[ℂ] H) τ.vector=0
  exact sub_eq_zero.mpr he

theorem norm₂_limit_star {I : Type*} {l : Filter I} (a : I → M.toStarSubalgebra)
    (b : M.toStarSubalgebra) (h : Tendsto (fun i => τ.norm₂ (a i-b)) l (𝓝 0)) :
    Tendsto (fun i => τ.norm₂ (star (a i)-star b)) l (𝓝 0) := by
  simpa only [← star_sub,τ.norm₂_star] using h

theorem norm₂_limit_mul {I : Type*} {l : Filter I}
    (a c : I → M.toStarSubalgebra) (b d : M.toStarSubalgebra)
    (C : ℝ) (hC : ∀ i, ‖a i‖ ≤ C)
    (ha : Tendsto (fun i => τ.norm₂ (a i-b)) l (𝓝 0))
    (hc : Tendsto (fun i => τ.norm₂ (c i-d)) l (𝓝 0)) :
    Tendsto (fun i => τ.norm₂ (a i*c i-b*d)) l (𝓝 0) := by
  have hbound : ∀ i, τ.norm₂ (a i*c i-b*d) ≤ C*τ.norm₂ (c i-d)+τ.norm₂ (a i-b)*‖d‖ := by
    intro i
    have he : a i*c i-b*d = a i*(c i-d)+(a i-b)*d := by
      simp only [mul_sub,sub_mul]; abel
    rw [he]
    exact (τ.norm₂_add_le _ _).trans (add_le_add
      ((τ.norm₂_mul_le _ _).trans (mul_le_mul_of_nonneg_right (hC i) (norm_nonneg _)))
      (τ.norm₂_mul_le_right _ _))
  apply squeeze_zero (fun i => norm_nonneg _) hbound
  simpa using (hc.const_mul C).add (ha.mul_const ‖d‖)

theorem norm₂_limit_trace {I : Type*} {l : Filter I}
    (a : I → M.toStarSubalgebra) (b : M.toStarSubalgebra)
    (h : Tendsto (fun i => τ.norm₂ (a i-b)) l (𝓝 0)) :
    Tendsto (fun i => τ.trace (a i)) l (𝓝 (τ.trace b)) := by
  apply (tendsto_iff_norm_sub_tendsto_zero).mpr
  apply squeeze_zero (fun i => norm_nonneg _)
    (fun i => ?_) h
  rw [← trace_sub]
  exact τ.norm_trace_le_norm₂ _

end Tracial

/-! A separating vector for a concrete von Neumann algebra is cyclic for its
commutant. The original representation need not be cyclic for the algebra. -/
section
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]

def vectorOrbitMap (S : StarSubalgebra ℂ (H →L[ℂ] H)) (ξ : H) : S →ₗ[ℂ] H where
  toFun a := a.val ξ
  map_add' _ _ := rfl
  map_smul' _ _ := rfl

def vectorOrbitClosure (S : StarSubalgebra ℂ (H →L[ℂ] H)) (ξ : H) : Submodule ℂ H :=
  (vectorOrbitMap S ξ).range.topologicalClosure

instance vectorOrbitClosure_complete (S : StarSubalgebra ℂ (H →L[ℂ] H)) (ξ : H) :
    CompleteSpace (vectorOrbitClosure S ξ) :=
  inferInstanceAs (CompleteSpace (vectorOrbitMap S ξ).range.topologicalClosure)

theorem mem_vectorOrbitClosure (S : StarSubalgebra ℂ (H →L[ℂ] H)) (ξ : H) :
    ξ ∈ vectorOrbitClosure S ξ :=
  Submodule.le_topologicalClosure _ ⟨1,rfl⟩

theorem vectorOrbitClosure_invariant (S : StarSubalgebra ℂ (H →L[ℂ] H)) (ξ : H)
    (a : S) {y : H} (hy : y ∈ vectorOrbitClosure S ξ) : a.val y ∈ vectorOrbitClosure S ξ := by
  have h : (vectorOrbitMap S ξ).range.topologicalClosure ≤
      (vectorOrbitClosure S ξ).comap a.val.toLinearMap := by
    apply Submodule.topologicalClosure_minimal
    · rintro z ⟨b,rfl⟩
      exact Submodule.le_topologicalClosure _ ⟨a*b,rfl⟩
    · exact (Submodule.isClosed_topologicalClosure _).preimage a.val.continuous
  exact h hy

theorem vectorOrbitProjection_commutes (S : StarSubalgebra ℂ (H →L[ℂ] H)) (ξ : H)
    (a : S) : (vectorOrbitClosure S ξ).starProjection*a.val=
      a.val*(vectorOrbitClosure S ξ).starProjection := by
  apply starProjection_commutes_of_invariant
  · exact fun y hy => vectorOrbitClosure_invariant S ξ a hy
  · exact fun y hy => vectorOrbitClosure_invariant S ξ (star a) hy

theorem commutant_orbitClosure_eq_top (M : VonNeumannAlgebra H) (ξ : H)
    (hξ : ∀ a : M.toStarSubalgebra,a.val ξ=0 → a=0) :
    vectorOrbitClosure M.commutant.toStarSubalgebra ξ=⊤ := by
  let K := vectorOrbitClosure M.commutant.toStarSubalgebra ξ
  have hQ : K.starProjection ∈ M := by
    have h : K.starProjection ∈ M.commutant.commutant := by
      apply VonNeumannAlgebra.mem_commutant_iff.mpr
      intro a ha
      exact (vectorOrbitProjection_commutes M.commutant.toStarSubalgebra ξ ⟨a,ha⟩).symm
    simpa only [M.commutant_commutant] using h
  have he : (1-K.starProjection) ξ=0 := by
    change ξ-K.starProjection ξ=0
    rw [Submodule.starProjection_eq_self_iff.mpr (mem_vectorOrbitClosure _ _),sub_self]
  have hzero := hξ ⟨1-K.starProjection,M.toStarSubalgebra.sub_mem M.one_mem hQ⟩ he
  have h1 : K.starProjection=1 := by
    have h := congrArg Subtype.val hzero
    exact (sub_eq_zero.mp h).symm
  apply top_unique
  intro y _
  apply Submodule.starProjection_eq_self_iff.mp
  rw [show (vectorOrbitClosure M.commutant.toStarSubalgebra ξ).starProjection=1 from h1]
  rfl

theorem commutant_orbit_dense (M : VonNeumannAlgebra H) (ξ : H)
    (hξ : ∀ a : M.toStarSubalgebra,a.val ξ=0 → a=0) :
    Dense (Set.range (fun a : M.commutant.toStarSubalgebra => a.val ξ)) := by
  intro y
  have h : y ∈ vectorOrbitClosure M.commutant.toStarSubalgebra ξ := by
    rw [commutant_orbitClosure_eq_top M ξ hξ]
    trivial
  exact h

end

/-! Restrict the faithful trace representation to the closure of M times its
trace vector. This constructs a faithful, isometric, cyclic representation on
an actual Hilbert subspace. No Banach predual or WOT-closed image is asserted. -/
namespace Tracial

section
open UltrapowerBridge
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
variable {M : VonNeumannAlgebra H} (τ : TracialRepresentation M)

abbrev traceCyclicSpace := vectorOrbitClosure M.toStarSubalgebra τ.vector

def traceCyclicVector : traceCyclicSpace τ := ⟨τ.vector,mem_vectorOrbitClosure _ _⟩

def traceCyclicOperator (a : M.toStarSubalgebra) :
    traceCyclicSpace τ →L[ℂ] traceCyclicSpace τ :=
  a.val.restrict (fun _ hy => vectorOrbitClosure_invariant _ _ a hy)

@[simp] theorem traceCyclicOperator_apply (a : M.toStarSubalgebra) (y : traceCyclicSpace τ) :
    (traceCyclicOperator τ a y).val=a.val y.val := rfl

theorem traceCyclicOperator_star (a : M.toStarSubalgebra) :
    traceCyclicOperator τ (star a)=star (traceCyclicOperator τ a) := by
  apply ContinuousLinearMap.ext
  intro x
  apply ext_inner_left ℂ
  intro y
  change inner ℂ y.val (star a.val x.val)=inner ℂ y (star (traceCyclicOperator τ a) x)
  simp only [ContinuousLinearMap.star_eq_adjoint,ContinuousLinearMap.adjoint_inner_right]
  rfl

def traceCyclicHom : M.toStarSubalgebra →⋆ₐ[ℂ]
    (traceCyclicSpace τ →L[ℂ] traceCyclicSpace τ) where
  toFun := traceCyclicOperator τ
  map_one' := by ext y; rfl
  map_mul' _ _ := by ext y; rfl
  map_zero' := by ext y; rfl
  map_add' _ _ := by ext y; rfl
  commutes' _ := by ext y; rfl
  map_star' := traceCyclicOperator_star τ

theorem traceCyclicHom_injective : Function.Injective (traceCyclicHom τ) := by
  intro a b hab
  apply sub_eq_zero.mp
  apply τ.separating
  have h := congrArg (fun T : traceCyclicSpace τ →L[ℂ] traceCyclicSpace τ =>
    (T (traceCyclicVector τ)).val) hab
  exact sub_eq_zero.mpr h

theorem traceCyclicHom_norm (a : M.toStarSubalgebra) : ‖traceCyclicHom τ a‖=‖a‖ := by
  let : CStarAlgebra M.toStarSubalgebra := {}
  let : CStarAlgebra (traceCyclicSpace τ →L[ℂ] traceCyclicSpace τ) := {}
  exact NonUnitalStarAlgHom.norm_map (traceCyclicHom τ) (traceCyclicHom_injective τ) a

theorem traceCyclicHom_isometry : Isometry (traceCyclicHom τ) :=
  AddMonoidHomClass.isometry_of_norm (traceCyclicHom τ) (traceCyclicHom_norm τ)

theorem traceCyclicVector_norm : ‖traceCyclicVector τ‖=1 := τ.norm_vector

theorem traceCyclicHom_trace (a : M.toStarSubalgebra) :
    inner ℂ (traceCyclicVector τ) (traceCyclicHom τ a (traceCyclicVector τ))=τ.trace a := rfl

theorem traceCyclicHom_norm₂ (a : M.toStarSubalgebra) :
    ‖traceCyclicHom τ a (traceCyclicVector τ)‖=τ.norm₂ a := rfl

theorem traceCyclicHom_cyclic :
    Dense (Set.range (fun a : M.toStarSubalgebra => traceCyclicHom τ a (traceCyclicVector τ))) := by
  intro y
  apply Metric.mem_closure_iff.mpr
  intro ε hε
  obtain ⟨z,⟨a,rfl⟩,ha⟩ := Metric.mem_closure_iff.mp y.property ε hε
  exact ⟨traceCyclicHom τ a (traceCyclicVector τ),⟨a,rfl⟩,ha⟩

theorem traceCyclicSpace_separable
    (hsep : TopologicalSpace.IsSeparable
      (Set.range (fun a : M.toStarSubalgebra => a.val τ.vector))) :
    TopologicalSpace.SeparableSpace (traceCyclicSpace τ) :=
  hsep.closure.separableSpace

end

/-! The faithful cyclic restriction sends actual algebra balls to compact
weak-operator balls. On each bounded ball it is a closed topological embedding. -/
open scoped Topology
open UltrapowerBridge
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
variable {M : VonNeumannAlgebra H} (τ : TracialRepresentation M)

def cyclicBallMap (C : ℝ) (T : weakOperatorBall M C) :
    traceCyclicSpace τ →WOT[ℂ] traceCyclicSpace τ :=
  ContinuousLinearMapWOT.ofCLM (traceCyclicHom τ ⟨T.val.toCLM,T.property.1⟩)

theorem cyclicBallMap_continuous (C : ℝ) : Continuous (cyclicBallMap τ C) := by
  apply ContinuousLinearMapWOT.continuous_iff.mpr
  intro x y
  change Continuous (fun T : weakOperatorBall M C => inner ℂ y.val (T.val x.val))
  exact ContinuousLinearMapWOT.continuous_inner_apply continuous_subtype_val x.val y.val

theorem cyclicBallMap_injective (C : ℝ) : Function.Injective (cyclicBallMap τ C) := by
  intro a b hab
  have he : traceCyclicHom τ ⟨a.val.toCLM,a.property.1⟩=
      traceCyclicHom τ ⟨b.val.toCLM,b.property.1⟩ :=
    congrArg ContinuousLinearMapWOT.toCLM hab
  have he' := congrArg Subtype.val (traceCyclicHom_injective τ he)
  apply Subtype.ext
  exact congrArg ContinuousLinearMapWOT.ofCLM he'

theorem cyclicBallMap_isClosedEmbedding (C : ℝ) :
    Topology.IsClosedEmbedding (cyclicBallMap τ C) := by
  let : CompactSpace (weakOperatorBall M C) :=
    isCompact_iff_compactSpace.mp (weakOperatorBall_isCompact M C)
  exact (cyclicBallMap_continuous τ C).isClosedEmbedding (cyclicBallMap_injective τ C)

def cyclicWeakOperatorBall (C : ℝ) : Set (traceCyclicSpace τ →WOT[ℂ] traceCyclicSpace τ) :=
  {T | ∃ a : M.toStarSubalgebra,‖a‖ ≤ C ∧ ContinuousLinearMapWOT.ofCLM (traceCyclicHom τ a)=T}

theorem cyclicWeakOperatorBall_eq_range (C : ℝ) :
    cyclicWeakOperatorBall τ C=Set.range (cyclicBallMap τ C) := by
  ext T
  constructor
  · rintro ⟨a,ha,rfl⟩
    exact ⟨⟨ContinuousLinearMapWOT.ofCLM a.val,⟨a.property,ha⟩⟩,rfl⟩
  · rintro ⟨a,rfl⟩
    exact ⟨⟨a.val.toCLM,a.property.1⟩,a.property.2,rfl⟩

theorem cyclicWeakOperatorBall_isCompact (C : ℝ) : IsCompact (cyclicWeakOperatorBall τ C) := by
  let : CompactSpace (weakOperatorBall M C) :=
    isCompact_iff_compactSpace.mp (weakOperatorBall_isCompact M C)
  rw [cyclicWeakOperatorBall_eq_range]
  exact isCompact_range (cyclicBallMap_continuous τ C)

theorem cyclicWeakOperatorBall_isClosed (C : ℝ) : IsClosed (cyclicWeakOperatorBall τ C) :=
  (cyclicWeakOperatorBall_isCompact τ C).isClosed

end Tracial

/-! A real linear functional continuous for bounded trace-two-norm sequences
is WOT continuous on the unit ball. Closed convex slices of the trace-vector
ball supply the conversion; no identification of the two topologies is assumed. -/
open Filter
open scoped Topology
namespace Tracial
open UltrapowerBridge
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
variable {M : VonNeumannAlgebra H} (τ : TracialRepresentation M)

theorem traceVector_weak_continuous (C : ℝ) :
    Continuous (fun T : weakOperatorBall M C => toWeakSpace ℂ H (T.val τ.vector)) := by
  apply WeakBilin.continuous_of_continuous_eval
  intro f
  exact (ContinuousLinearMapWOT.continuous_dual_apply τ.vector f).comp continuous_subtype_val

def traceFunctionalSlice (f : M.toStarSubalgebra →L[ℝ] ℝ) (r : ℝ) : Set H :=
  {ξ | ∃ a : M.toStarSubalgebra, ‖a‖ ≤ 1 ∧ f a ≤ r ∧ a.val τ.vector = ξ}

theorem traceFunctionalSlice_convex (f : M.toStarSubalgebra →L[ℝ] ℝ) (r : ℝ) :
    Convex ℝ (traceFunctionalSlice τ f r) := by
  intro x hx y hy s t hs ht hst
  obtain ⟨a, ha, hfa, rfl⟩ := hx
  obtain ⟨b, hb, hfb, rfl⟩ := hy
  refine ⟨s • a + t • b, ?_, ?_, rfl⟩
  · calc
      ‖s • a + t • b‖ ≤ ‖s • a‖ + ‖t • b‖ := norm_add_le _ _
      _ = s * ‖a‖ + t * ‖b‖ := by
        rw [norm_smul, norm_smul, Real.norm_of_nonneg hs, Real.norm_of_nonneg ht]
      _ ≤ s * 1 + t * 1 := _root_.add_le_add
        (mul_le_mul_of_nonneg_left ha hs) (mul_le_mul_of_nonneg_left hb ht)
      _ = 1 := by simpa only [mul_one] using hst
  · rw [map_add, map_smul, map_smul, smul_eq_mul, smul_eq_mul]
    calc
      s*f a+t*f b ≤ s*r+t*r := _root_.add_le_add
        (mul_le_mul_of_nonneg_left hfa hs) (mul_le_mul_of_nonneg_left hfb ht)
      _ = r := by rw [← add_mul, hst, one_mul]

theorem traceFunctionalSlice_isClosed (f : M.toStarSubalgebra →L[ℝ] ℝ)
    (hf : ∀ (a : ℕ → M.toStarSubalgebra) (b : M.toStarSubalgebra),
      (∀ n, ‖a n‖ ≤ 1) → Tendsto (fun n => τ.norm₂ (a n-b)) atTop (𝓝 0) →
      Tendsto (fun n => f (a n)) atTop (𝓝 (f b))) (r : ℝ) :
    IsClosed (traceFunctionalSlice τ f r) := by
  apply IsSeqClosed.isClosed
  intro ξ z hz hlim
  choose a ha hfa he using hz
  have hva : Tendsto (fun n => (a n).val τ.vector) atTop (𝓝 z) := by
    simpa only [he] using hlim
  obtain ⟨b, hb, hbe⟩ := bounded_trace_limit τ a 1 zero_le_one ha z hva
  refine ⟨b, hb, ?_, hbe⟩
  apply le_of_tendsto (hf a b ha ?_) (Eventually.of_forall hfa)
  apply (tendsto_vector_iff_norm₂ τ atTop a b).mp
  rwa [hbe]

theorem unitBall_functional_sublevel_isClosed (f : M.toStarSubalgebra →L[ℝ] ℝ)
    (hf : ∀ (a : ℕ → M.toStarSubalgebra) (b : M.toStarSubalgebra),
      (∀ n, ‖a n‖ ≤ 1) → Tendsto (fun n => τ.norm₂ (a n-b)) atTop (𝓝 0) →
      Tendsto (fun n => f (a n)) atTop (𝓝 (f b))) (r : ℝ) :
    IsClosed {T : weakOperatorBall M 1 | f ⟨T.val.toCLM, T.property.1⟩ ≤ r} := by
  have hclosed := traceFunctionalSlice_isClosed τ f hf r
  have hconvex := traceFunctionalSlice_convex τ f r
  have hw : IsClosed ((toWeakSpace ℂ H) '' traceFunctionalSlice τ f r) := by
    rw [← closure_eq_iff_isClosed, ← hconvex.toWeakSpace_closure ℂ, hclosed.closure_eq]
  have hpre := hw.preimage (traceVector_weak_continuous τ 1)
  convert hpre using 1
  ext T
  constructor
  · intro h
    exact ⟨T.val τ.vector, ⟨⟨T.val.toCLM, T.property.1⟩, T.property.2, h, rfl⟩, rfl⟩
  · rintro ⟨ξ, ⟨a, _, hfa, ha⟩, he⟩
    have hv : a.val τ.vector = T.val τ.vector := ha.trans ((toWeakSpace ℂ H).injective he)
    have heq : a = (⟨T.val.toCLM, T.property.1⟩ : M.toStarSubalgebra) := by
      apply sub_eq_zero.mp
      apply τ.separating
      change a.val τ.vector - T.val τ.vector = 0
      exact sub_eq_zero.mpr hv
    change f ⟨T.val.toCLM, T.property.1⟩ ≤ r
    simpa only [heq] using hfa

theorem unitBall_functional_continuous_of_norm₂ (f : M.toStarSubalgebra →L[ℝ] ℝ)
    (hf : ∀ (a : ℕ → M.toStarSubalgebra) (b : M.toStarSubalgebra),
      (∀ n, ‖a n‖ ≤ 1) → Tendsto (fun n => τ.norm₂ (a n-b)) atTop (𝓝 0) →
      Tendsto (fun n => f (a n)) atTop (𝓝 (f b))) :
    Continuous (fun T : weakOperatorBall M 1 => f ⟨T.val.toCLM, T.property.1⟩) := by
  apply continuous_iff_lower_upperSemicontinuous.mpr
  constructor
  · exact lowerSemicontinuous_iff_isClosed_preimage.mpr
      (unitBall_functional_sublevel_isClosed τ f hf)
  · apply upperSemicontinuous_iff_isClosed_preimage.mpr
    intro r
    have hn := unitBall_functional_sublevel_isClosed τ (-f)
      (fun a b ha ht => (hf a b ha ht).neg) (-r)
    change IsClosed {T : weakOperatorBall M 1 | r ≤ f ⟨T.val.toCLM, T.property.1⟩}
    simpa only [neg_apply, neg_le_neg_iff] using hn

end Tracial

/-! On a uniformly operator-norm bounded family in a faithfully vector-traced
von Neumann algebra, L2 convergence implies strong and adjoint-strong convergence
in the original Hilbert space. -/
open Filter
open scoped Topology
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]

omit [CompleteSpace H] in
theorem bounded_operators_tendsto_of_dense {I : Type*} {l : Filter I}
    (a : I → H →L[ℂ] H) (b : H →L[ℂ] H) (C : ℝ)
    (ha : ∀ i,‖a i‖ ≤ C) (s : Set H) (hs : Dense s)
    (hlim : ∀ y ∈ s,Tendsto (fun i => a i y) l (𝓝 (b y))) :
    ∀ y : H,Tendsto (fun i => a i y) l (𝓝 (b y)) := by
  intro y
  apply Metric.tendsto_nhds.mpr
  intro ε hε
  let D := |C|+‖b‖+1
  have hD : 0 < D := by dsimp [D]; positivity
  obtain ⟨z,hz,hyz⟩ := Metric.mem_closure_iff.mp (hs y) (ε/(2*D)) (by positivity)
  have hzlim := Metric.tendsto_nhds.mp (hlim z hz) (ε/2) (by positivity)
  filter_upwards [hzlim] with i hi
  have he : a i y-b y=(a i-b) (y-z)+(a i z-b z) := by
    simp only [sub_apply,map_sub]; abel
  have hn : ‖a i-b‖ ≤ D := by
    calc
      ‖a i-b‖ ≤ ‖a i‖+‖b‖ := norm_sub_le _ _
      _ ≤ |C|+‖b‖ := _root_.add_le_add ((ha i).trans (le_abs_self C)) le_rfl
      _ ≤ D := le_add_of_nonneg_right zero_le_one
  have hd : D*‖y-z‖ < ε/2 := by
    have h := (mul_lt_mul_iff_right₀ hD).mpr hyz
    rw [dist_eq_norm] at h
    have heq : D*(ε/(2*D))=ε/2 := by field_simp
    rw [heq] at h
    exact h
  rw [dist_eq_norm,he]
  have hbound := ((a i-b).le_opNorm (y-z)).trans
    (mul_le_mul_of_nonneg_right hn (norm_nonneg _))
  have hi' : ‖a i z-b z‖ < ε/2 := by simpa only [dist_eq_norm] using hi
  exact (norm_add_le _ _).trans_lt (by linarith)
namespace Tracial
open UltrapowerBridge
variable {M : VonNeumannAlgebra H} (τ : TracialRepresentation M)
theorem norm₂_limit_strong {I : Type*} {l : Filter I}
    (a : I → M.toStarSubalgebra) (b : M.toStarSubalgebra) (C : ℝ)
    (ha : ∀ i,‖a i‖ ≤ C)
    (hlim : Tendsto (fun i => τ.norm₂ (a i-b)) l (𝓝 0)) :
    ∀ y : H,Tendsto (fun i => (a i).val y) l (𝓝 (b.val y)) := by
  apply bounded_operators_tendsto_of_dense (fun i => (a i).val) b.val C ha
    (Set.range (fun c : M.commutant.toStarSubalgebra => c.val τ.vector))
    (commutant_orbit_dense M τ.vector τ.separating)
  rintro _ ⟨c,rfl⟩
  have he (d : M.toStarSubalgebra) : d.val (c.val τ.vector)=c.val (d.val τ.vector) :=
    congrArg (fun T : H →L[ℂ] H => T τ.vector)
      (VonNeumannAlgebra.mem_commutant_iff.mp c.property d.val d.property)
  simp only [he]
  exact c.val.continuous.continuousAt.tendsto.comp
    ((tendsto_vector_iff_norm₂ τ l a b).mpr hlim)
theorem norm₂_limit_strong_star {I : Type*} {l : Filter I}
    (a : I → M.toStarSubalgebra) (b : M.toStarSubalgebra) (C : ℝ)
    (ha : ∀ i,‖a i‖ ≤ C)
    (hlim : Tendsto (fun i => τ.norm₂ (a i-b)) l (𝓝 0)) :
    ∀ y : H,Tendsto (fun i => (star (a i)).val y) l (𝓝 ((star b).val y)) :=
  norm₂_limit_strong τ (fun i => star (a i)) (star b) C
    (fun i => by
      change ‖star ((a i).val)‖ ≤ C
      rw [ContinuousLinearMap.star_eq_adjoint,ContinuousLinearMap.adjoint.norm_map]
      exact ha i) (norm₂_limit_star τ a b hlim)
theorem bounded_strong_iff_norm₂ {I : Type*} {l : Filter I}
    (a : I → M.toStarSubalgebra) (b : M.toStarSubalgebra) (C : ℝ)
    (ha : ∀ i,‖a i‖ ≤ C) :
    (∀ y : H,Tendsto (fun i => (a i).val y) l (𝓝 (b.val y))) ↔
      Tendsto (fun i => τ.norm₂ (a i-b)) l (𝓝 0) :=
  ⟨fun h => (tendsto_vector_iff_norm₂ τ l a b).mp (h τ.vector),
    norm₂_limit_strong τ a b C ha⟩
end Tracial
end Generator
end
end OAI

	import Mathlib.Analysis.CStarAlgebra.Hom
	import Mathlib.Analysis.LocallyConvex.WeakSpace
	import Mathlib.Topology.Semicontinuity.Basic
	import OAI.Analysis.FactorGeneration.Bicommutant
	import OAI.Analysis.FactorGeneration.CornerTrace

	namespace OAI

	noncomputable section

	namespace Generator

	/-! Multiplication and adjoint limits in L2, with the necessary uniform
	operator norm bounds stated explicitly. -/
	namespace Tracial
	open Filter
	open scoped Topology
	open UltrapowerBridge
	variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
	variable {M : VonNeumannAlgebra H} (τ : TracialRepresentation M)

	theorem tendsto_vector_iff_norm₂ {I : Type*} (l : Filter I)
	(a : I → M.toStarSubalgebra) (b : M.toStarSubalgebra) :
	Tendsto (fun i => (a i : H →L[ℂ] H) τ.vector) l (𝓝 ((b : H →L[ℂ] H) τ.vector)) ↔
	Tendsto (fun i => τ.norm₂ (a i-b)) l (𝓝 0) := by
	simpa only [dist_eq_norm,TracialRepresentation.norm₂_sub] using
	(tendsto_iff_dist_tendsto_zero (f := fun i => (a i : H →L[ℂ] H) τ.vector)
	(a := (b : H →L[ℂ] H) τ.vector))

	theorem norm₂_limit_unique {I : Type*} {l : Filter I} [l.NeBot]
	(a : I → M.toStarSubalgebra) {b c : M.toStarSubalgebra}
	(hb : Tendsto (fun i => τ.norm₂ (a i-b)) l (𝓝 0))
	(hc : Tendsto (fun i => τ.norm₂ (a i-c)) l (𝓝 0)) : b=c := by
	have he := tendsto_nhds_unique ((tendsto_vector_iff_norm₂ τ l a b).mpr hb)
	((tendsto_vector_iff_norm₂ τ l a c).mpr hc)
	apply sub_eq_zero.mp
	apply τ.separating
	change (b : H →L[ℂ] H) τ.vector-(c : H →L[ℂ] H) τ.vector=0
	exact sub_eq_zero.mpr he

	theorem norm₂_limit_star {I : Type*} {l : Filter I} (a : I → M.toStarSubalgebra)
	(b : M.toStarSubalgebra) (h : Tendsto (fun i => τ.norm₂ (a i-b)) l (𝓝 0)) :
	Tendsto (fun i => τ.norm₂ (star (a i)-star b)) l (𝓝 0) := by
	simpa only [← star_sub,τ.norm₂_star] using h

	theorem norm₂_limit_mul {I : Type*} {l : Filter I}
	(a c : I → M.toStarSubalgebra) (b d : M.toStarSubalgebra)
	(C : ℝ) (hC : ∀ i, ‖a i‖ ≤ C)
	(ha : Tendsto (fun i => τ.norm₂ (a i-b)) l (𝓝 0))
	(hc : Tendsto (fun i => τ.norm₂ (c i-d)) l (𝓝 0)) :
	Tendsto (fun i => τ.norm₂ (a ic i-bd)) l (𝓝 0) := by
	have hbound : ∀ i, τ.norm₂ (a ic i-bd) ≤ Cτ.norm₂ (c i-d)+τ.norm₂ (a i-b)‖d‖ := by
	intro i
	have he : a ic i-bd = a i(c i-d)+(a i-b)d := by
	simp only [mul_sub,sub_mul]; abel
	rw [he]
	exact (τ.norm₂_add_le _ _).trans (add_le_add
	((τ.norm₂_mul_le _ _).trans (mul_le_mul_of_nonneg_right (hC i) (norm_nonneg _)))
	(τ.norm₂_mul_le_right _ _))
	apply squeeze_zero (fun i => norm_nonneg _) hbound
	simpa using (hc.const_mul C).add (ha.mul_const ‖d‖)

	theorem norm₂_limit_trace {I : Type*} {l : Filter I}
	(a : I → M.toStarSubalgebra) (b : M.toStarSubalgebra)
	(h : Tendsto (fun i => τ.norm₂ (a i-b)) l (𝓝 0)) :
	Tendsto (fun i => τ.trace (a i)) l (𝓝 (τ.trace b)) := by
	apply (tendsto_iff_norm_sub_tendsto_zero).mpr
	apply squeeze_zero (fun i => norm_nonneg _)
	(fun i => ?_) h
	rw [← trace_sub]
	exact τ.norm_trace_le_norm₂ _

	end Tracial

	/-! A separating vector for a concrete von Neumann algebra is cyclic for its
	commutant. The original representation need not be cyclic for the algebra. -/
	section
	variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]

	def vectorOrbitMap (S : StarSubalgebra ℂ (H →L[ℂ] H)) (ξ : H) : S →ₗ[ℂ] H where
	toFun a := a.val ξ
	map_add' _ _ := rfl
	map_smul' _ _ := rfl

	def vectorOrbitClosure (S : StarSubalgebra ℂ (H →L[ℂ] H)) (ξ : H) : Submodule ℂ H :=
	(vectorOrbitMap S ξ).range.topologicalClosure

	instance vectorOrbitClosure_complete (S : StarSubalgebra ℂ (H →L[ℂ] H)) (ξ : H) :
	CompleteSpace (vectorOrbitClosure S ξ) :=
	inferInstanceAs (CompleteSpace (vectorOrbitMap S ξ).range.topologicalClosure)

	theorem mem_vectorOrbitClosure (S : StarSubalgebra ℂ (H →L[ℂ] H)) (ξ : H) :
	ξ ∈ vectorOrbitClosure S ξ :=
	Submodule.le_topologicalClosure _ ⟨1,rfl⟩

	theorem vectorOrbitClosure_invariant (S : StarSubalgebra ℂ (H →L[ℂ] H)) (ξ : H)
	(a : S) {y : H} (hy : y ∈ vectorOrbitClosure S ξ) : a.val y ∈ vectorOrbitClosure S ξ := by
	have h : (vectorOrbitMap S ξ).range.topologicalClosure ≤
	(vectorOrbitClosure S ξ).comap a.val.toLinearMap := by
	apply Submodule.topologicalClosure_minimal
	· rintro z ⟨b,rfl⟩
	exact Submodule.le_topologicalClosure _ ⟨a*b,rfl⟩
	· exact (Submodule.isClosed_topologicalClosure _).preimage a.val.continuous
	exact h hy

	theorem vectorOrbitProjection_commutes (S : StarSubalgebra ℂ (H →L[ℂ] H)) (ξ : H)
	(a : S) : (vectorOrbitClosure S ξ).starProjection*a.val=
	a.val*(vectorOrbitClosure S ξ).starProjection := by
	apply starProjection_commutes_of_invariant
	· exact fun y hy => vectorOrbitClosure_invariant S ξ a hy
	· exact fun y hy => vectorOrbitClosure_invariant S ξ (star a) hy

	theorem commutant_orbitClosure_eq_top (M : VonNeumannAlgebra H) (ξ : H)
	(hξ : ∀ a : M.toStarSubalgebra,a.val ξ=0 → a=0) :
	vectorOrbitClosure M.commutant.toStarSubalgebra ξ=⊤ := by
	let K := vectorOrbitClosure M.commutant.toStarSubalgebra ξ
	have hQ : K.starProjection ∈ M := by
	have h : K.starProjection ∈ M.commutant.commutant := by
	apply VonNeumannAlgebra.mem_commutant_iff.mpr
	intro a ha
	exact (vectorOrbitProjection_commutes M.commutant.toStarSubalgebra ξ ⟨a,ha⟩).symm
	simpa only [M.commutant_commutant] using h
	have he : (1-K.starProjection) ξ=0 := by
	change ξ-K.starProjection ξ=0
	rw [Submodule.starProjection_eq_self_iff.mpr (mem_vectorOrbitClosure _ _),sub_self]
	have hzero := hξ ⟨1-K.starProjection,M.toStarSubalgebra.sub_mem M.one_mem hQ⟩ he
	have h1 : K.starProjection=1 := by
	have h := congrArg Subtype.val hzero
	exact (sub_eq_zero.mp h).symm
	apply top_unique
	intro y _
	apply Submodule.starProjection_eq_self_iff.mp
	rw [show (vectorOrbitClosure M.commutant.toStarSubalgebra ξ).starProjection=1 from h1]
	rfl

	theorem commutant_orbit_dense (M : VonNeumannAlgebra H) (ξ : H)
	(hξ : ∀ a : M.toStarSubalgebra,a.val ξ=0 → a=0) :
	Dense (Set.range (fun a : M.commutant.toStarSubalgebra => a.val ξ)) := by
	intro y
	have h : y ∈ vectorOrbitClosure M.commutant.toStarSubalgebra ξ := by
	rw [commutant_orbitClosure_eq_top M ξ hξ]
	trivial
	exact h

	end

	/-! Restrict the faithful trace representation to the closure of M times its
	trace vector. This constructs a faithful, isometric, cyclic representation on
	an actual Hilbert subspace. No Banach predual or WOT-closed image is asserted. -/
	namespace Tracial

	section
	open UltrapowerBridge
	variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
	variable {M : VonNeumannAlgebra H} (τ : TracialRepresentation M)

	abbrev traceCyclicSpace := vectorOrbitClosure M.toStarSubalgebra τ.vector

	def traceCyclicVector : traceCyclicSpace τ := ⟨τ.vector,mem_vectorOrbitClosure _ _⟩

	def traceCyclicOperator (a : M.toStarSubalgebra) :
	traceCyclicSpace τ →L[ℂ] traceCyclicSpace τ :=
	a.val.restrict (fun _ hy => vectorOrbitClosure_invariant _ _ a hy)

	@[simp] theorem traceCyclicOperator_apply (a : M.toStarSubalgebra) (y : traceCyclicSpace τ) :
	(traceCyclicOperator τ a y).val=a.val y.val := rfl

	theorem traceCyclicOperator_star (a : M.toStarSubalgebra) :
	traceCyclicOperator τ (star a)=star (traceCyclicOperator τ a) := by
	apply ContinuousLinearMap.ext
	intro x
	apply ext_inner_left ℂ
	intro y
	change inner ℂ y.val (star a.val x.val)=inner ℂ y (star (traceCyclicOperator τ a) x)
	simp only [ContinuousLinearMap.star_eq_adjoint,ContinuousLinearMap.adjoint_inner_right]
	rfl

	def traceCyclicHom : M.toStarSubalgebra →⋆ₐ[ℂ]
	(traceCyclicSpace τ →L[ℂ] traceCyclicSpace τ) where
	toFun := traceCyclicOperator τ
	map_one' := by ext y; rfl
	map_mul' _ _ := by ext y; rfl
	map_zero' := by ext y; rfl
	map_add' _ _ := by ext y; rfl
	commutes' _ := by ext y; rfl
	map_star' := traceCyclicOperator_star τ

	theorem traceCyclicHom_injective : Function.Injective (traceCyclicHom τ) := by
	intro a b hab
	apply sub_eq_zero.mp
	apply τ.separating
	have h := congrArg (fun T : traceCyclicSpace τ →L[ℂ] traceCyclicSpace τ =>
	(T (traceCyclicVector τ)).val) hab
	exact sub_eq_zero.mpr h

	theorem traceCyclicHom_norm (a : M.toStarSubalgebra) : ‖traceCyclicHom τ a‖=‖a‖ := by
	let : CStarAlgebra M.toStarSubalgebra := {}
	let : CStarAlgebra (traceCyclicSpace τ →L[ℂ] traceCyclicSpace τ) := {}
	exact NonUnitalStarAlgHom.norm_map (traceCyclicHom τ) (traceCyclicHom_injective τ) a

	theorem traceCyclicHom_isometry : Isometry (traceCyclicHom τ) :=
	AddMonoidHomClass.isometry_of_norm (traceCyclicHom τ) (traceCyclicHom_norm τ)

	theorem traceCyclicVector_norm : ‖traceCyclicVector τ‖=1 := τ.norm_vector

	theorem traceCyclicHom_trace (a : M.toStarSubalgebra) :
	inner ℂ (traceCyclicVector τ) (traceCyclicHom τ a (traceCyclicVector τ))=τ.trace a := rfl

	theorem traceCyclicHom_norm₂ (a : M.toStarSubalgebra) :
	‖traceCyclicHom τ a (traceCyclicVector τ)‖=τ.norm₂ a := rfl

	theorem traceCyclicHom_cyclic :
	Dense (Set.range (fun a : M.toStarSubalgebra => traceCyclicHom τ a (traceCyclicVector τ))) := by
	intro y
	apply Metric.mem_closure_iff.mpr
	intro ε hε
	obtain ⟨z,⟨a,rfl⟩,ha⟩ := Metric.mem_closure_iff.mp y.property ε hε
	exact ⟨traceCyclicHom τ a (traceCyclicVector τ),⟨a,rfl⟩,ha⟩

	theorem traceCyclicSpace_separable
	(hsep : TopologicalSpace.IsSeparable
	(Set.range (fun a : M.toStarSubalgebra => a.val τ.vector))) :
	TopologicalSpace.SeparableSpace (traceCyclicSpace τ) :=
	hsep.closure.separableSpace

	end

	/-! The faithful cyclic restriction sends actual algebra balls to compact
	weak-operator balls. On each bounded ball it is a closed topological embedding. -/
	open scoped Topology
	open UltrapowerBridge
	variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
	variable {M : VonNeumannAlgebra H} (τ : TracialRepresentation M)

	def cyclicBallMap (C : ℝ) (T : weakOperatorBall M C) :
	traceCyclicSpace τ →WOT[ℂ] traceCyclicSpace τ :=
	ContinuousLinearMapWOT.ofCLM (traceCyclicHom τ ⟨T.val.toCLM,T.property.1⟩)

	theorem cyclicBallMap_continuous (C : ℝ) : Continuous (cyclicBallMap τ C) := by
	apply ContinuousLinearMapWOT.continuous_iff.mpr
	intro x y
	change Continuous (fun T : weakOperatorBall M C => inner ℂ y.val (T.val x.val))
	exact ContinuousLinearMapWOT.continuous_inner_apply continuous_subtype_val x.val y.val

	theorem cyclicBallMap_injective (C : ℝ) : Function.Injective (cyclicBallMap τ C) := by
	intro a b hab
	have he : traceCyclicHom τ ⟨a.val.toCLM,a.property.1⟩=
	traceCyclicHom τ ⟨b.val.toCLM,b.property.1⟩ :=
	congrArg ContinuousLinearMapWOT.toCLM hab
	have he' := congrArg Subtype.val (traceCyclicHom_injective τ he)
	apply Subtype.ext
	exact congrArg ContinuousLinearMapWOT.ofCLM he'

	theorem cyclicBallMap_isClosedEmbedding (C : ℝ) :
	Topology.IsClosedEmbedding (cyclicBallMap τ C) := by
	let : CompactSpace (weakOperatorBall M C) :=
	isCompact_iff_compactSpace.mp (weakOperatorBall_isCompact M C)
	exact (cyclicBallMap_continuous τ C).isClosedEmbedding (cyclicBallMap_injective τ C)

	def cyclicWeakOperatorBall (C : ℝ) : Set (traceCyclicSpace τ →WOT[ℂ] traceCyclicSpace τ) :=
	{T \| ∃ a : M.toStarSubalgebra,‖a‖ ≤ C ∧ ContinuousLinearMapWOT.ofCLM (traceCyclicHom τ a)=T}

	theorem cyclicWeakOperatorBall_eq_range (C : ℝ) :
	cyclicWeakOperatorBall τ C=Set.range (cyclicBallMap τ C) := by
	ext T
	constructor
	· rintro ⟨a,ha,rfl⟩
	exact ⟨⟨ContinuousLinearMapWOT.ofCLM a.val,⟨a.property,ha⟩⟩,rfl⟩
	· rintro ⟨a,rfl⟩
	exact ⟨⟨a.val.toCLM,a.property.1⟩,a.property.2,rfl⟩

	theorem cyclicWeakOperatorBall_isCompact (C : ℝ) : IsCompact (cyclicWeakOperatorBall τ C) := by
	let : CompactSpace (weakOperatorBall M C) :=
	isCompact_iff_compactSpace.mp (weakOperatorBall_isCompact M C)
	rw [cyclicWeakOperatorBall_eq_range]
	exact isCompact_range (cyclicBallMap_continuous τ C)

	theorem cyclicWeakOperatorBall_isClosed (C : ℝ) : IsClosed (cyclicWeakOperatorBall τ C) :=
	(cyclicWeakOperatorBall_isCompact τ C).isClosed

	end Tracial

	/-! A real linear functional continuous for bounded trace-two-norm sequences
	is WOT continuous on the unit ball. Closed convex slices of the trace-vector
	ball supply the conversion; no identification of the two topologies is assumed. -/
	open Filter
	open scoped Topology
	namespace Tracial
	open UltrapowerBridge
	variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
	variable {M : VonNeumannAlgebra H} (τ : TracialRepresentation M)

	theorem traceVector_weak_continuous (C : ℝ) :
	Continuous (fun T : weakOperatorBall M C => toWeakSpace ℂ H (T.val τ.vector)) := by
	apply WeakBilin.continuous_of_continuous_eval
	intro f
	exact (ContinuousLinearMapWOT.continuous_dual_apply τ.vector f).comp continuous_subtype_val

	def traceFunctionalSlice (f : M.toStarSubalgebra →L[ℝ] ℝ) (r : ℝ) : Set H :=
	{ξ \| ∃ a : M.toStarSubalgebra, ‖a‖ ≤ 1 ∧ f a ≤ r ∧ a.val τ.vector = ξ}

	theorem traceFunctionalSlice_convex (f : M.toStarSubalgebra →L[ℝ] ℝ) (r : ℝ) :
	Convex ℝ (traceFunctionalSlice τ f r) := by
	intro x hx y hy s t hs ht hst
	obtain ⟨a, ha, hfa, rfl⟩ := hx
	obtain ⟨b, hb, hfb, rfl⟩ := hy
	refine ⟨s • a + t • b, ?_, ?_, rfl⟩
	· calc
	‖s • a + t • b‖ ≤ ‖s • a‖ + ‖t • b‖ := norm_add_le _ _
	_ = s * ‖a‖ + t * ‖b‖ := by
	rw [norm_smul, norm_smul, Real.norm_of_nonneg hs, Real.norm_of_nonneg ht]
	_ ≤ s * 1 + t * 1 := _root_.add_le_add
	(mul_le_mul_of_nonneg_left ha hs) (mul_le_mul_of_nonneg_left hb ht)
	_ = 1 := by simpa only [mul_one] using hst
	· rw [map_add, map_smul, map_smul, smul_eq_mul, smul_eq_mul]
	calc
	sf a+tf b ≤ sr+tr := _root_.add_le_add
	(mul_le_mul_of_nonneg_left hfa hs) (mul_le_mul_of_nonneg_left hfb ht)
	_ = r := by rw [← add_mul, hst, one_mul]

	theorem traceFunctionalSlice_isClosed (f : M.toStarSubalgebra →L[ℝ] ℝ)
	(hf : ∀ (a : ℕ → M.toStarSubalgebra) (b : M.toStarSubalgebra),
	(∀ n, ‖a n‖ ≤ 1) → Tendsto (fun n => τ.norm₂ (a n-b)) atTop (𝓝 0) →
	Tendsto (fun n => f (a n)) atTop (𝓝 (f b))) (r : ℝ) :
	IsClosed (traceFunctionalSlice τ f r) := by
	apply IsSeqClosed.isClosed
	intro ξ z hz hlim
	choose a ha hfa he using hz
	have hva : Tendsto (fun n => (a n).val τ.vector) atTop (𝓝 z) := by
	simpa only [he] using hlim
	obtain ⟨b, hb, hbe⟩ := bounded_trace_limit τ a 1 zero_le_one ha z hva
	refine ⟨b, hb, ?_, hbe⟩
	apply le_of_tendsto (hf a b ha ?_) (Eventually.of_forall hfa)
	apply (tendsto_vector_iff_norm₂ τ atTop a b).mp
	rwa [hbe]

	theorem unitBall_functional_sublevel_isClosed (f : M.toStarSubalgebra →L[ℝ] ℝ)
	(hf : ∀ (a : ℕ → M.toStarSubalgebra) (b : M.toStarSubalgebra),
	(∀ n, ‖a n‖ ≤ 1) → Tendsto (fun n => τ.norm₂ (a n-b)) atTop (𝓝 0) →
	Tendsto (fun n => f (a n)) atTop (𝓝 (f b))) (r : ℝ) :
	IsClosed {T : weakOperatorBall M 1 \| f ⟨T.val.toCLM, T.property.1⟩ ≤ r} := by
	have hclosed := traceFunctionalSlice_isClosed τ f hf r
	have hconvex := traceFunctionalSlice_convex τ f r
	have hw : IsClosed ((toWeakSpace ℂ H) '' traceFunctionalSlice τ f r) := by
	rw [← closure_eq_iff_isClosed, ← hconvex.toWeakSpace_closure ℂ, hclosed.closure_eq]
	have hpre := hw.preimage (traceVector_weak_continuous τ 1)
	convert hpre using 1
	ext T
	constructor
	· intro h
	exact ⟨T.val τ.vector, ⟨⟨T.val.toCLM, T.property.1⟩, T.property.2, h, rfl⟩, rfl⟩
	· rintro ⟨ξ, ⟨a, _, hfa, ha⟩, he⟩
	have hv : a.val τ.vector = T.val τ.vector := ha.trans ((toWeakSpace ℂ H).injective he)
	have heq : a = (⟨T.val.toCLM, T.property.1⟩ : M.toStarSubalgebra) := by
	apply sub_eq_zero.mp
	apply τ.separating
	change a.val τ.vector - T.val τ.vector = 0
	exact sub_eq_zero.mpr hv
	change f ⟨T.val.toCLM, T.property.1⟩ ≤ r
	simpa only [heq] using hfa

	theorem unitBall_functional_continuous_of_norm₂ (f : M.toStarSubalgebra →L[ℝ] ℝ)
	(hf : ∀ (a : ℕ → M.toStarSubalgebra) (b : M.toStarSubalgebra),
	(∀ n, ‖a n‖ ≤ 1) → Tendsto (fun n => τ.norm₂ (a n-b)) atTop (𝓝 0) →
	Tendsto (fun n => f (a n)) atTop (𝓝 (f b))) :
	Continuous (fun T : weakOperatorBall M 1 => f ⟨T.val.toCLM, T.property.1⟩) := by
	apply continuous_iff_lower_upperSemicontinuous.mpr
	constructor
	· exact lowerSemicontinuous_iff_isClosed_preimage.mpr
	(unitBall_functional_sublevel_isClosed τ f hf)
	· apply upperSemicontinuous_iff_isClosed_preimage.mpr
	intro r
	have hn := unitBall_functional_sublevel_isClosed τ (-f)
	(fun a b ha ht => (hf a b ha ht).neg) (-r)
	change IsClosed {T : weakOperatorBall M 1 \| r ≤ f ⟨T.val.toCLM, T.property.1⟩}
	simpa only [neg_apply, neg_le_neg_iff] using hn

	end Tracial

	/-! On a uniformly operator-norm bounded family in a faithfully vector-traced
	von Neumann algebra, L2 convergence implies strong and adjoint-strong convergence
	in the original Hilbert space. -/
	open Filter
	open scoped Topology
	variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]

	omit [CompleteSpace H] in
	theorem bounded_operators_tendsto_of_dense {I : Type*} {l : Filter I}
	(a : I → H →L[ℂ] H) (b : H →L[ℂ] H) (C : ℝ)
	(ha : ∀ i,‖a i‖ ≤ C) (s : Set H) (hs : Dense s)
	(hlim : ∀ y ∈ s,Tendsto (fun i => a i y) l (𝓝 (b y))) :
	∀ y : H,Tendsto (fun i => a i y) l (𝓝 (b y)) := by
	intro y
	apply Metric.tendsto_nhds.mpr
	intro ε hε
	let D := \|C\|+‖b‖+1
	have hD : 0 < D := by dsimp [D]; positivity
	obtain ⟨z,hz,hyz⟩ := Metric.mem_closure_iff.mp (hs y) (ε/(2*D)) (by positivity)
	have hzlim := Metric.tendsto_nhds.mp (hlim z hz) (ε/2) (by positivity)
	filter_upwards [hzlim] with i hi
	have he : a i y-b y=(a i-b) (y-z)+(a i z-b z) := by
	simp only [sub_apply,map_sub]; abel
	have hn : ‖a i-b‖ ≤ D := by
	calc
	‖a i-b‖ ≤ ‖a i‖+‖b‖ := norm_sub_le _ _
	_ ≤ \|C\|+‖b‖ := _root_.add_le_add ((ha i).trans (le_abs_self C)) le_rfl
	_ ≤ D := le_add_of_nonneg_right zero_le_one
	have hd : D*‖y-z‖ < ε/2 := by
	have h := (mul_lt_mul_iff_right₀ hD).mpr hyz
	rw [dist_eq_norm] at h
	have heq : D(ε/(2D))=ε/2 := by field_simp
	rw [heq] at h
	exact h
	rw [dist_eq_norm,he]
	have hbound := ((a i-b).le_opNorm (y-z)).trans
	(mul_le_mul_of_nonneg_right hn (norm_nonneg _))
	have hi' : ‖a i z-b z‖ < ε/2 := by simpa only [dist_eq_norm] using hi
	exact (norm_add_le _ _).trans_lt (by linarith)

	namespace Tracial
	open UltrapowerBridge
	variable {M : VonNeumannAlgebra H} (τ : TracialRepresentation M)

	theorem norm₂_limit_strong {I : Type*} {l : Filter I}
	(a : I → M.toStarSubalgebra) (b : M.toStarSubalgebra) (C : ℝ)
	(ha : ∀ i,‖a i‖ ≤ C)
	(hlim : Tendsto (fun i => τ.norm₂ (a i-b)) l (𝓝 0)) :
	∀ y : H,Tendsto (fun i => (a i).val y) l (𝓝 (b.val y)) := by
	apply bounded_operators_tendsto_of_dense (fun i => (a i).val) b.val C ha
	(Set.range (fun c : M.commutant.toStarSubalgebra => c.val τ.vector))
	(commutant_orbit_dense M τ.vector τ.separating)
	rintro _ ⟨c,rfl⟩
	have he (d : M.toStarSubalgebra) : d.val (c.val τ.vector)=c.val (d.val τ.vector) :=
	congrArg (fun T : H →L[ℂ] H => T τ.vector)
	(VonNeumannAlgebra.mem_commutant_iff.mp c.property d.val d.property)
	simp only [he]
	exact c.val.continuous.continuousAt.tendsto.comp
	((tendsto_vector_iff_norm₂ τ l a b).mpr hlim)

	theorem norm₂_limit_strong_star {I : Type*} {l : Filter I}
	(a : I → M.toStarSubalgebra) (b : M.toStarSubalgebra) (C : ℝ)
	(ha : ∀ i,‖a i‖ ≤ C)
	(hlim : Tendsto (fun i => τ.norm₂ (a i-b)) l (𝓝 0)) :
	∀ y : H,Tendsto (fun i => (star (a i)).val y) l (𝓝 ((star b).val y)) :=
	norm₂_limit_strong τ (fun i => star (a i)) (star b) C
	(fun i => by
	change ‖star ((a i).val)‖ ≤ C
	rw [ContinuousLinearMap.star_eq_adjoint,ContinuousLinearMap.adjoint.norm_map]
	exact ha i) (norm₂_limit_star τ a b hlim)

	theorem bounded_strong_iff_norm₂ {I : Type*} {l : Filter I}
	(a : I → M.toStarSubalgebra) (b : M.toStarSubalgebra) (C : ℝ)
	(ha : ∀ i,‖a i‖ ≤ C) :
	(∀ y : H,Tendsto (fun i => (a i).val y) l (𝓝 (b.val y))) ↔
	Tendsto (fun i => τ.norm₂ (a i-b)) l (𝓝 0) :=
	⟨fun h => (tendsto_vector_iff_norm₂ τ l a b).mp (h τ.vector),
	norm₂_limit_strong τ a b C ha⟩

	end Tracial

	end Generator

	end

	end OAI