| import OAI.Analysis.FiniteFactor.Diffuse.OperatorSupport |
| import Mathlib.Analysis.CStarAlgebra.Basic |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace Nonhyperlinear.Diffuseness |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
|
|
| lemma projection_vector_trace (v:H) {p:H→L[ℂ]H} (hp:IsStarProjection p) : |
| inner ℂ v (p v)=(‖p v‖:ℂ)^2 := by |
| have he : (star p)*(p)=p := by rw [hp.isSelfAdjoint.star_eq,hp.isIdempotentElem.eq] |
| calc |
| inner ℂ v (p v)=inner ℂ v ((star p) (p v)) := congrArg (fun A:H→L[ℂ]H=>inner ℂ v (A v)) he.symm |
| _ = _ := by |
| rw [ContinuousLinearMap.star_eq_adjoint,ContinuousLinearMap.adjoint_inner_right, |
| inner_self_eq_norm_sq_to_K] |
| norm_cast |
|
|
| lemma projection_vector_trace_nonneg (v:H) {p:H→L[ℂ]H} (hp:IsStarProjection p) : |
| 0≤(inner ℂ v (p v)).re := by |
| rw [projection_vector_trace v hp] |
| simp only [←Complex.ofReal_pow,Complex.ofReal_re] |
| positivity |
|
|
| lemma projection_difference {p q:H→L[ℂ]H} (hp:IsStarProjection p) (hq:IsStarProjection q) |
| (hqp:q*p=q) : IsStarProjection (p-q) := by |
| have hpq:p*q=q := by |
| have hh:=congrArg star hqp |
| simpa only [star_mul,hp.isSelfAdjoint.star_eq,hq.isSelfAdjoint.star_eq] using hh |
| apply isStarProjection_iff'.mpr |
| constructor |
| · calc |
| (p-q)*(p-q)=p*p-p*q-q*p+q*q := by noncomm_ring |
| _ = p-q := by rw [hp.isIdempotentElem.eq,hq.isIdempotentElem.eq,hpq,hqp]; abel |
| · simp only [star_sub,hp.isSelfAdjoint.star_eq,hq.isSelfAdjoint.star_eq] |
|
|
| lemma projection_vector_trace_mono (v:H) {p q:H→L[ℂ]H} |
| (hp:IsStarProjection p) (hq:IsStarProjection q) (hqp:q*p=q) : |
| (inner ℂ v (q v)).re≤(inner ℂ v (p v)).re := by |
| have hh:=projection_vector_trace_nonneg v (projection_difference hp hq hqp) |
| change 0≤(inner ℂ v (p v-q v)).re at hh |
| rw [inner_sub_right,Complex.sub_re] at hh |
| linarith |
|
|
| end Nonhyperlinear.Diffuseness |
|
|
| end |
|
|
| end OAI |
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|
