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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 | import OAI.Analysis.IrrationalRotation.Model.OperatorProperties
import OAI.Analysis.IrrationalRotation.Rotation.ProjectionDiagonalPropagation
namespace OAI
noncomputable section
open scoped InnerProductSpace
namespace Invariant233
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H]
[CompleteSpace H]
theorem strict_positive_trace_of_nonzero_projection
(M : VonNeumannAlgebra H) (ξ : H) (htrace : IsFaithfulTracialVector M ξ)
(P : H →L[ℂ] H) (hPm : P ∈ M) (hP : IsStarProjection P) (hP0 : P ≠ 0) :
0 < (vectorTrace ξ P).re := by
have hnonneg := (InvariantProjection.projection_re_inner_bounds hP ξ).1
apply lt_of_le_of_ne hnonneg
intro heq
have hzero : P ξ = 0 :=
(InvariantProjection.projection_re_inner_eq_zero_iff hP ξ).mp heq.symm
apply hP0
exact htrace.2.2 P hPm (ContinuousLinearMap.IsPositive.of_isStarProjection hP)
(by simp only [vectorTrace, hzero, inner_zero_right])
theorem strict_trace_bounds_of_nontrivial_projection
(M : VonNeumannAlgebra H) (ξ : H) (htrace : IsFaithfulTracialVector M ξ)
(P : H →L[ℂ] H) (hPm : P ∈ M) (hP : IsStarProjection P)
(hP0 : P ≠ 0) (hP1 : P ≠ 1) :
0 < (vectorTrace ξ P).re ∧ (vectorTrace ξ P).re < 1 := by
refine ⟨strict_positive_trace_of_nonzero_projection M ξ htrace P hPm hP hP0, ?_⟩
have hcomp := strict_positive_trace_of_nonzero_projection M ξ htrace
(1 - P) (M.sub_mem M.one_mem hPm) hP.one_sub
(sub_ne_zero.mpr (Ne.symm hP1))
have hself : (inner ℂ ξ ξ).re = 1 := by
calc
(inner ℂ ξ ξ).re = ‖ξ‖ ^ 2 := inner_self_eq_norm_sq (𝕜 := ℂ) ξ
_ = 1 := by rw [htrace.1]; norm_num
simp only [vectorTrace, sub_apply,
one_apply_eq_self, inner_sub_right, Complex.sub_re,
hself] at hcomp ⊢
exact sub_pos.mp hcomp
end Invariant233
end
end OAI
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