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9eeb9c7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 | import OAI.Analysis.IrrationalRotation.Model.NormalAffiliation
import OAI.Analysis.IrrationalRotation.Operators.AffiliatedTransport
import Mathlib.Analysis.CStarAlgebra.Unitary.Span
namespace OAI
noncomputable section
namespace Invariant233
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
[CompleteSpace E]
theorem IsAffiliated.graph_mem_of_mem_commutant
{M : VonNeumannAlgebra E} {H : E →ₗ.[ℂ] E} (hH : IsAffiliated M H)
{W : E →L[ℂ] E} (hW : W ∈ M.commutant) (x : H.domain) :
(W (x : E), W (H x)) ∈ H.graph := by
let S : Submodule ℂ M.commutant := {
carrier := {a | ∀ z : H.domain,
((a : E →L[ℂ] E) (z : E), (a : E →L[ℂ] E) (H z)) ∈ H.graph}
zero_mem' := by
intro z
simp
add_mem' := by
intro a b ha hb z
change ((a : E →L[ℂ] E) z + (b : E →L[ℂ] E) z,
(a : E →L[ℂ] E) (H z) + (b : E →L[ℂ] E) (H z)) ∈ H.graph
exact H.graph.add_mem (ha z) (hb z)
smul_mem' := by
intro c a ha z
simpa only [SetLike.val_smul, smul_apply, Prod.smul_mk]
using H.graph.smul_mem c (ha z) }
have hspan : Submodule.span ℂ (unitary M.commutant : Set M.commutant) ≤ S := by
apply Submodule.span_le.mpr
intro U hU z
let U' : unitary (E →L[ℂ] E) :=
⟨(U : E →L[ℂ] E), Unitary.map_mem M.commutant.toStarSubalgebra.subtype hU⟩
obtain ⟨hD, hC⟩ := hH U' U.property
exact H.mem_graph_iff.mpr ⟨⟨(U' : E →L[ℂ] E) z, hD z⟩, rfl, hC z⟩
rw [CStarAlgebra.span_unitary] at hspan
exact hspan (Submodule.mem_top : (⟨W, hW⟩ : M.commutant) ∈ ⊤) x
theorem IsAffiliated.commutes_of_mem_commutant
{M : VonNeumannAlgebra E} {H : E →ₗ.[ℂ] E} (hH : IsAffiliated M H)
{W : E →L[ℂ] E} (hW : W ∈ M.commutant) :
∃ hD : ∀ x : H.domain, W (x : E) ∈ H.domain,
∀ x : H.domain, H ⟨W (x : E), hD x⟩ = W (H x) := by
let hD : ∀ x : H.domain, W (x : E) ∈ H.domain := fun x =>
H.mem_domain_of_mem_graph (hH.graph_mem_of_mem_commutant hW x)
refine ⟨hD, fun x => ?_⟩
exact H.mem_graph_snd_inj (H.mem_graph ⟨W x, hD x⟩)
(hH.graph_mem_of_mem_commutant hW x) rfl
theorem IsAffiliated.transported_projection_mem
{M : VonNeumannAlgebra E} {H : E →ₗ.[ℂ] E} (hH : IsAffiliated M H)
{p : E →L[ℂ] E} (hpM : p ∈ M) (hp : IsStarProjection p) :
(InvariantRotation.AffiliatedTransport.transported H p.range).starProjection ∈ M := by
apply (VonNeumannAlgebra.IsStarProjection.mem_iff
isStarProjection_starProjection M).mpr
intro W hW
rw [Submodule.range_starProjection]
apply (Module.End.mem_invtSubmodule_iff_forall_mem_of_mem _).mpr
obtain ⟨hD, hC⟩ := hH.commutes_of_mem_commutant hW
have hpW := (VonNeumannAlgebra.IsStarProjection.mem_iff hp M).mp hpM W hW
exact InvariantRotation.AffiliatedTransport.transported_invariant_of_commutes H p.range W
((Module.End.mem_invtSubmodule_iff_forall_mem_of_mem _).mp hpW)
(fun x hx => hD ⟨x, hx⟩) hC
theorem IsAffiliated.transported_projection_invariant
{M : VonNeumannAlgebra E} {H : E →ₗ.[ℂ] E} (hH : IsAffiliated M H)
(A B : E →L[ℂ] E)
(hD : ∀ x : H.domain, A (x : E) ∈ H.domain)
(hI : ∀ x : H.domain, H ⟨A (x : E), hD x⟩ = B (H x))
{p : E →L[ℂ] E} (hpM : p ∈ M) (hp : IsStarProjection p)
(hinv : (1 - p) * A * p = 0) :
let Q := (InvariantRotation.AffiliatedTransport.transported H p.range).starProjection
Q ∈ M ∧ IsStarProjection Q ∧ (1 - Q) * B * Q = 0 := by
dsimp only
refine ⟨hH.transported_projection_mem hpM hp, isStarProjection_starProjection, ?_⟩
have hpap : p * A * p = A * p := by
rw [sub_mul, one_mul, sub_mul, sub_eq_zero] at hinv
exact hinv.symm
have hpA : p.range ∈ Module.End.invtSubmodule A := by
apply (ContinuousLinearMap.IsIdempotentElem.range_mem_invtSubmodule_iff
hp.isIdempotentElem).mpr
exact hpap
have hQB := InvariantRotation.AffiliatedTransport.transported_invariant H p.range A B
((Module.End.mem_invtSubmodule_iff_forall_mem_of_mem _).mp hpA)
(fun x hx => hD ⟨x, hx⟩) hI
have hQinv :
(InvariantRotation.AffiliatedTransport.transported H p.range).starProjection.range ∈
Module.End.invtSubmodule B := by
rw [Submodule.range_starProjection]
exact (Module.End.mem_invtSubmodule_iff_forall_mem_of_mem _).mpr hQB
have hQBQ := (ContinuousLinearMap.IsIdempotentElem.range_mem_invtSubmodule_iff
isStarProjection_starProjection.isIdempotentElem).mp hQinv
simp only [← ContinuousLinearMap.mul_def] at hQBQ
rw [sub_mul, one_mul, sub_mul, mul_assoc, hQBQ, sub_self]
end Invariant233
end
end OAI
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