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ddc84c5 | 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 | 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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