| import Mathlib |
| import OAI.Analysis.BrownMeasure.Modulus |
| import OAI.Analysis.BrownMeasure.Spectrum |
| import OAI.Analysis.FiniteFactor.CrossedProduct.Trace |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory Filter InnerProductSpace |
| open scoped ComplexInnerProductSpace Topology |
| namespace FiniteFactor |
|
|
| def sourcePolarUnitary (θ : SourceCoordinate → ℝ) (n : ℤ) : unitary ProductOp := |
| ⟨sourceTransport θ 0 n,by |
| constructor <;> simp only [sourceTransport_star,neg_zero,sourceTransport_mul, |
| zero_add,neg_add_cancel,add_neg_cancel,sourceTransport_zero]⟩ |
|
|
| def sourcePolarAutomorphism (θ : SourceCoordinate → ℝ) (n : ℤ) : ProductOp≃⋆ₐ[ℂ]ProductOp := |
| Unitary.conjStarAlgAut ℂ ProductOp (sourcePolarUnitary θ n) |
|
|
| lemma sourcePolarAutomorphism_apply (θ : SourceCoordinate → ℝ) (n : ℤ) (A : ProductOp) : |
| sourcePolarAutomorphism θ n A=sourceTransport θ 0 n*A*sourceTransport θ 0 (-n) := by |
| unfold sourcePolarAutomorphism |
| rw [Unitary.conjStarAlgAut_apply] |
| change sourceTransport θ 0 n*A*star (sourceTransport θ 0 n)=_ |
| rw [sourceTransport_star,neg_zero] |
|
|
| lemma sourcePolarAutomorphism_zero (θ : SourceCoordinate → ℝ) (A : ProductOp) : |
| sourcePolarAutomorphism θ 0 A=A := by |
| simp [sourcePolarAutomorphism_apply,sourceTransport_zero] |
|
|
| lemma sourcePolarAutomorphism_add (θ : SourceCoordinate → ℝ) (n m : ℤ) (A : ProductOp) : |
| sourcePolarAutomorphism θ (n+m) A=sourcePolarAutomorphism θ n (sourcePolarAutomorphism θ m A) := by |
| simp only [sourcePolarAutomorphism_apply] |
| rw [show sourceTransport θ 0 (n+m)=sourceTransport θ 0 n*sourceTransport θ 0 m by |
| rw [sourceTransport_mul,zero_add]] |
| rw [show sourceTransport θ 0 (-(n+m))=sourceTransport θ 0 (-m)*sourceTransport θ 0 (-n) by |
| rw [sourceTransport_mul,zero_add]; congr 1; omega] |
| noncomm_ring |
|
|
| lemma sourcePolarAutomorphism_one (θ : SourceCoordinate → ℝ) (A : ProductOp) : |
| sourcePolarAutomorphism θ 1 A=sourceShift SourceCoordinate*A*star (sourceShift SourceCoordinate) := by |
| rw [sourcePolarAutomorphism_apply,sourceShift_eq_transport θ,sourceTransport_star,neg_zero] |
|
|
| lemma sourcePolarAutomorphism_family (θ : SourceCoordinate → ℝ) (hθ : SourceAngles θ) (n m : ℤ) : |
| sourcePolarAutomorphism θ n (sourcePolarFamily θ hθ m)=sourcePolarFamily θ hθ (n+m) := by |
| rw [← sourceTransport_conjugate_pi θ hθ m,← sourceTransport_conjugate_pi θ hθ (n+m)] |
| simpa only [sourcePolarAutomorphism_apply] using (sourcePolarAutomorphism_add θ n m (sourcePi θ (sourceWeightBase θ hθ))).symm |
|
|
| lemma sourcePolarTransport_mem (θ : SourceCoordinate → ℝ) (n : ℤ) : |
| sourceTransport θ 0 n∈sourceAlgebra θ := by |
| induction n using Int.induction_on with |
| | zero => rw [sourceTransport_zero]; exact one_mem _ |
| | @succ n hn => |
| rw [show sourceTransport θ 0 (n+1)=sourceTransport θ 0 n*sourceTransport θ 0 1 by |
| rw [sourceTransport_mul,zero_add]] |
| exact mul_mem hn (by rw [← sourceShift_eq_transport θ]; exact sourceShift_mem θ) |
| | @pred n hn => |
| rw [show sourceTransport θ 0 (-(n:ℤ)-1)=sourceTransport θ 0 (-(n:ℤ))*sourceTransport θ 0 (-1) by |
| rw [sourceTransport_mul,zero_add]; congr 1] |
| apply mul_mem hn |
| have he : sourceTransport θ 0 (-1)=star (sourceShift SourceCoordinate) := by |
| rw [sourceShift_eq_transport θ,sourceTransport_star,neg_zero] |
| rw [he] |
| exact star_mem (sourceShift_mem θ) |
|
|
| lemma sourcePolarAutomorphism_trace (θ : SourceCoordinate → ℝ) (n : ℤ) {A : ProductOp} |
| (hA : A∈sourceAlgebra θ) : |
| sourceVacuumFunctional (sourcePolarAutomorphism θ n A)=sourceVacuumFunctional A := by |
| rw [sourcePolarAutomorphism_apply, |
| sourceVacuumFunctional_tracial θ (mul_mem (sourcePolarTransport_mem θ n) hA) |
| (sourcePolarTransport_mem θ (-n)),← mul_assoc,sourceTransport_mul] |
| simp only [zero_add,neg_add_cancel,sourceTransport_zero,one_mul] |
|
|
| lemma sourceWeight_injective (θ : SourceCoordinate → ℝ) (hθ : SourceAngles θ) : |
| Function.Injective (sourcePi θ (sourceWeightBase θ hθ)) := by |
| intro u v he |
| apply sourceOperator_injective θ sourceLogWeight hθ.nonneg sourceLogWeight_nonneg |
| change sourceShift SourceCoordinate (sourcePi θ (sourceWeightBase θ hθ) u)= |
| sourceShift SourceCoordinate (sourcePi θ (sourceWeightBase θ hθ) v) |
| rw [he] |
|
|
| lemma sourceOperator_spectrum (θ : SourceCoordinate → ℝ) (hθ : SourceAngles θ) : |
| spectrum ℂ (sourceOperator θ sourceLogWeight hθ.nonneg sourceLogWeight_nonneg)={0} := |
| spectrum_eq_zero_of_root_norm_tendsto _ (staged_sourceOperator_properties θ hθ).2.2.2 |
| end FiniteFactor |
| end |
|
|
| end OAI |
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|
