lean/OAI/Analysis/FactorGeneration/OrbitMinima.lean

import Mathlib.Analysis.CStarAlgebra.Unitary.Span
import OAI.Analysis.FactorGeneration.CornerTrace

namespace OAI

noncomputable section

namespace Generator.Tracial

/-! The least-norm unitary-orbit point is scalar when the coordinate inclusion has scalar commutant. -/
section
open scoped Topology
open UltrapowerBridge
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
variable {M : VonNeumannAlgebra H} (τ : TracialRepresentation M)

def subalgebraUnitaries (P : StarSubalgebra ℂ M.toStarSubalgebra) : Subgroup (unitary M.toStarSubalgebra) where
  carrier := {u | (u : M.toStarSubalgebra) ∈ P}
  one_mem' := P.one_mem
  mul_mem' hu hv := P.mul_mem hu hv
  inv_mem' hu := by
    change star _ ∈ P
    exact star_mem hu

def ambientUnitary (P : StarSubalgebra ℂ M.toStarSubalgebra) (u : unitary P) :
    subalgebraUnitaries P :=
  ⟨Unitary.map { toFun := P.subtype, map_one' := map_one _, map_mul' := map_mul _, map_star' := map_star _ } u, u.val.property⟩

theorem commutes_of_fixed_unitaries (P : StarSubalgebra ℂ M.toStarSubalgebra) [CompleteSpace P]
    (c : M.toStarSubalgebra)
    (hc : ∀ u : subalgebraUnitaries P, conjugateLinear u.val c = c) :
    ∀ a : P, (a : M.toStarSubalgebra)*c = c*(a : M.toStarSubalgebra) := by
  let : CStarAlgebra P := {}
  intro a
  obtain ⟨u, s, ha, _⟩ := CStarAlgebra.exists_sum_four_unitary a
  have hu (i : Fin 4) : ((u i).val : M.toStarSubalgebra)*c = c*((u i).val : M.toStarSubalgebra) := by
    have h := congrArg (fun z : M.toStarSubalgebra => z * ((u i).val : M.toStarSubalgebra)) (hc (ambientUnitary P (u i)))
    change (((u i).val : M.toStarSubalgebra)*c*star ((u i).val : M.toStarSubalgebra))*((u i).val : M.toStarSubalgebra) = _ at h
    have he : star ((u i).val : M.toStarSubalgebra)*((u i).val : M.toStarSubalgebra) = 1 :=
      congrArg Subtype.val (u i).property.1
    simpa only [mul_assoc, he, mul_one] using h
  change P.subtype a*c = c*P.subtype a
  rw [ha, map_sum, Finset.sum_mul, Finset.mul_sum]
  apply Finset.sum_congr rfl
  intro i _
  simp only [map_smul, smul_mul_assoc, mul_smul_comm]
  exact congrArg (fun z : M.toStarSubalgebra => s i • z) (hu i)

/-- Irreducibility is exactly the coordinate relative-commutant hypothesis.
Completeness of P is automatic for a von Neumann subalgebra, and is used only
to express its elements as sums of four unitaries. -/
theorem scalar_mem_traceOrbitHull (P : StarSubalgebra ℂ M.toStarSubalgebra) [CompleteSpace P]
    (hirr : ∀ c : M.toStarSubalgebra, (∀ a : P, (a : M.toStarSubalgebra)*c = c*(a : M.toStarSubalgebra)) →
      ∃ z : ℂ, c = z • (1 : M.toStarSubalgebra)) (b : M.toStarSubalgebra) :
    baseVector τ (τ.trace b • (1 : M.toStarSubalgebra)) ∈ traceOrbitHull τ (subalgebraUnitaries P) b := by
  obtain ⟨c, _, hc, _, hfixed⟩ := traceOrbitHull_minimum τ (subalgebraUnitaries P) b
  obtain ⟨z, rfl⟩ := hirr c (commutes_of_fixed_unitaries P c hfixed)
  have hz := traceOrbitHull_trace τ (subalgebraUnitaries P) b hc
  have he : z = τ.trace b := by
    change inner ℂ τ.vector (z • τ.vector) = _ at hz
    rw [inner_smul_right, inner_self_eq_norm_sq_to_K, τ.norm_vector] at hz
    simpa using hz
  simpa only [he] using hc

theorem trace_distance_le_unitary_displacement (P : StarSubalgebra ℂ M.toStarSubalgebra) [CompleteSpace P]
    (hirr : ∀ c : M.toStarSubalgebra, (∀ a : P, (a : M.toStarSubalgebra)*c = c*(a : M.toStarSubalgebra)) →
      ∃ z : ℂ, c = z • (1 : M.toStarSubalgebra))
    (b : M.toStarSubalgebra) (δ : ℝ)
    (hδ : ∀ u : subalgebraUnitaries P, τ.norm₂ (conjugateLinear u.val b-b) ≤ δ) :
    τ.norm₂ (b-τ.trace b • (1 : M.toStarSubalgebra)) ≤ δ := by
  have h := traceOrbitHull_distance τ (subalgebraUnitaries P) b δ hδ
    (scalar_mem_traceOrbitHull τ P hirr b)
  rw [norm_sub_rev] at h
  exact h

end

/-! Irreducibility excludes disconnected nonzero ambient projections.
This gives the projection part of irreducibility in every nonzero corner. -/
open scoped Topology
open UltrapowerBridge
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
variable {M : VonNeumannAlgebra H} (τ : TracialRepresentation M)
variable (P : StarSubalgebra ℂ M.toStarSubalgebra) [CompleteSpace P]
variable (hirr : ∀ c : M.toStarSubalgebra,
  (∀ a : P, (a : M.toStarSubalgebra)*c=c*(a : M.toStarSubalgebra)) →
  ∃ z : ℂ, c=z • (1 : M.toStarSubalgebra))

theorem baseVector_inner_eq_trace (a b : M.toStarSubalgebra) :
    inner ℂ (baseVector τ a) (baseVector τ b)=τ.trace (star a*b) := by
  change _ = inner ℂ τ.vector (star (a : H →L[ℂ] H) ((b : H →L[ℂ] H) τ.vector))
  rw [ContinuousLinearMap.star_eq_adjoint,ContinuousLinearMap.adjoint_inner_right]
  rfl

/-- A continuous trace test vanishing on an orbit vanishes on its closed
convex hull. -/
theorem orbitHull_trace_test_zero (G : Subgroup (unitary M.toStarSubalgebra))
    (b d : M.toStarSubalgebra)
    (hd : ∀ u : G, τ.trace (star d * conjugateLinear u.val b)=0)
    {ξ : H} (hξ : ξ ∈ traceOrbitHull τ G b) :
    inner ℂ (baseVector τ d) ξ=0 := by
  let K : Set H := {ξ | inner ℂ (baseVector τ d) ξ=0}
  have hc : IsClosed K := isClosed_eq (continuous_const.inner continuous_id) continuous_const
  have hv : Convex ℝ K := by
    intro x hx y hy s t hs ht hst
    change inner ℂ (baseVector τ d) (s • x+t • y)=0
    rw [inner_add_right,inner_smul_right_eq_smul,inner_smul_right_eq_smul]
    change s • inner ℂ (baseVector τ d) x+t • inner ℂ (baseVector τ d) y=0
    rw [show inner ℂ (baseVector τ d) x=0 from hx,
      show inner ℂ (baseVector τ d) y=0 from hy,smul_zero,smul_zero,add_zero]
  apply (closure_minimal (convexHull_min ?_ hv) hc) hξ
  rintro _ ⟨a,⟨u,rfl⟩,rfl⟩
  exact (baseVector_inner_eq_trace τ d (conjugateLinear u.val b)).trans (hd u)

include τ hirr in
/-- Nonzero ambient projections cannot have all their P matrix coefficients
zero in an irreducible inclusion. No comparison theorem is assumed. -/
theorem projection_eq_zero_or_eq_zero_of_no_links (r s : AlgebraProjection M)
    (hlink : ∀ a : P, s.val*(a : M.toStarSubalgebra)*r.val=0) :
    r.val=0 ∨ s.val=0 := by
  have htest : ∀ u : subalgebraUnitaries P,
      τ.trace (star s.val*conjugateLinear u.val r.val)=0 := by
    intro u
    change τ.trace (star s.val*((u.val : M.toStarSubalgebra)*r.val*
      star (u.val : M.toStarSubalgebra)))=0
    rw [s.property.isSelfAdjoint.star_eq,← mul_assoc,← mul_assoc,
      hlink ⟨u.val.val,u.property⟩,zero_mul]
    exact inner_zero_right _
  have hz := orbitHull_trace_test_zero τ (subalgebraUnitaries P) r.val s.val htest
    (scalar_mem_traceOrbitHull τ P hirr r.val)
  rw [map_smul,inner_smul_right,baseVector_inner_eq_trace,
    s.property.isSelfAdjoint.star_eq,mul_one] at hz
  rcases mul_eq_zero.mp hz with hr | hs
  · left
    by_contra hn
    have hp := projection_trace_pos τ r.val r.property hn
    rw [hr,Complex.zero_re] at hp
    exact (lt_irrefl 0 hp)
  · right
    by_contra hn
    have hp := projection_trace_pos τ s.val s.property hn
    rw [hs,Complex.zero_re] at hp
    exact (lt_irrefl 0 hp)

include τ hirr in
/-- Every projection commuting with pPp inside pMp is either zero or p. -/
theorem corner_commutant_projection_trivial (p r : AlgebraProjection M)
    (hrp : r ≤ p)
    (hr : ∀ a : P, Commute r.val (p.val*(a : M.toStarSubalgebra)*p.val)) :
    r.val=0 ∨ r.val=p.val := by
  let s : AlgebraProjection M := ⟨p.val-r.val,r.property.sub_of_mul_eq_right p.property hrp⟩
  have hlink (a : P) : s.val*(a : M.toStarSubalgebra)*r.val=0 := by
    have hl : p.val*r.val=r.val := hrp
    have hright := projection_mul_of_le hrp
    have he := congrArg (fun z : M.toStarSubalgebra => z*r.val) (hr a).eq
    simp only [mul_assoc,hl,r.property.isIdempotentElem.eq] at he
    have he' : r.val*((p.val*(a : M.toStarSubalgebra))*r.val)=
        (p.val*(a : M.toStarSubalgebra))*r.val := by
      simpa only [mul_assoc,hl,r.property.isIdempotentElem.eq] using he
    have he'' : r.val*(a : M.toStarSubalgebra)*r.val=
        p.val*(a : M.toStarSubalgebra)*r.val := by
      simpa only [← mul_assoc,hright] using he'
    change (p.val-r.val)*(a : M.toStarSubalgebra)*r.val=0
    rw [sub_mul,sub_mul,he'',sub_self]
  rcases projection_eq_zero_or_eq_zero_of_no_links τ P hirr r s hlink with h | h
  · exact Or.inl h
  · exact Or.inr (sub_eq_zero.mp h).symm

end Generator.Tracial

end

end OAI