| 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 |
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|
