File size: 2,580 Bytes
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 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 | import OAI.Analysis.FiniteFactor.Diffuse.OperatorMeanErgodic
import Mathlib.Analysis.VonNeumannAlgebra.Basic
namespace OAI
noncomputable section
open Filter
open scoped Topology
namespace Nonhyperlinear.Diffuseness
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
/-- The actual orthogonal projection onto the closed range of an operator. -/
def rangeSupport (T : H →L[ℂ] H) : H →L[ℂ] H :=
T.range.topologicalClosure.starProjection
lemma rangeSupport_projection (T : H →L[ℂ] H) : IsStarProjection (rangeSupport T) :=
isStarProjection_starProjection
lemma rangeSupport_range (T : H →L[ℂ] H) :
(rangeSupport T).range=T.range.topologicalClosure :=
Submodule.range_starProjection _
lemma rangeSupport_mul_self (T : H →L[ℂ] H) : rangeSupport T*T=T := by
ext x
apply Submodule.starProjection_eq_self_iff.mpr
exact Submodule.le_topologicalClosure _ ⟨x,rfl⟩
lemma rangeSupport_ne_zero {T : H →L[ℂ] H} (hT:T≠0) : rangeSupport T≠0 := by
intro h
have he:=rangeSupport_mul_self T
rw [h,zero_mul] at he
exact hT he.symm
/-- Closed range support belongs to the same concrete von Neumann algebra. -/
theorem rangeSupport_mem (M : VonNeumannAlgebra H) {T:H→L[ℂ]H} (hT:T∈M) :
rangeSupport T∈M := by
apply (VonNeumannAlgebra.IsStarProjection.mem_iff (rangeSupport_projection T) M).mpr
intro y hy
rw [rangeSupport_range]
apply Submodule.topologicalClosure_mem_invtSubmodule
intro x hx
obtain ⟨z,rfl⟩:=hx
refine ⟨y z,?_⟩
have hc := VonNeumannAlgebra.mem_commutant_iff.mp hy T hT
exact congrArg (fun A:H→L[ℂ]H=>A z) hc
lemma projection_mul_rangeSupport {p T:H→L[ℂ]H} (hp:IsStarProjection p) (h:p*T=T) :
p*rangeSupport T=rangeSupport T := by
have hs : Set.EqOn p id (T.range:Set H) := by
rintro _ ⟨x,rfl⟩
have hx : T x ∈ p.range := ⟨T x, congrArg (fun A:H→L[ℂ]H=>A x) h⟩
obtain ⟨y,hy⟩ := hx
change p (T x) = T x
rw [← hy]
exact congrArg (fun A:H→L[ℂ]H=>A y) hp.isIdempotentElem.eq
have hc : Set.EqOn p id (closure (T.range:Set H)) :=
hs.closure p.continuous continuous_id
ext x
exact hc (T.range.topologicalClosure.starProjection_apply_mem x)
lemma rangeSupport_mul_projection {p T:H→L[ℂ]H} (hp:IsStarProjection p) (h:p*T=T) :
rangeSupport T*p=rangeSupport T := by
have he:=congrArg star (projection_mul_rangeSupport hp h)
simpa only [star_mul,hp.isSelfAdjoint.star_eq,(rangeSupport_projection T).isSelfAdjoint.star_eq] using he
end Nonhyperlinear.Diffuseness
end
end OAI
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