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/- | ||
Copyright (c) 2024 Jireh Loreaux. All rights reserved. | ||
Released under Apache 2.0 license as described in the file LICENSE. | ||
Authors: Jireh Loreaux | ||
-/ | ||
import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | ||
import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | ||
import Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow | ||
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/-! # Positive contractions in a C⋆-algebra form an approximate unit | ||
This file will show (although it does not yet) that the collection of positive contractions (of norm | ||
strictly less than one) in a possibly non-unital C⋆-algebra form an approximate unit. The key step | ||
is to show that this set is directed using the continuous functional calculus applied with the | ||
functions `fun x : ℝ≥0, 1 - (1 + x)⁻¹` and `fun x : ℝ≥0, x * (1 - x)⁻¹`, which are inverses on | ||
the interval `{x : ℝ≥0 | x < 1}`. | ||
## Main declarations | ||
+ `CFC.monotoneOn_one_sub_one_add_inv`: the function `f := fun x : ℝ≥0, 1 - (1 + x)⁻¹` is | ||
*operator monotone* on `Set.Ici (0 : A)` (i.e., `cfcₙ f` is monotone on `{x : A | 0 ≤ x}`). | ||
+ `Set.InvOn.one_sub_one_add_inv`: the functions `f := fun x : ℝ≥0, 1 - (1 + x)⁻¹` and | ||
`g := fun x : ℝ≥0, x * (1 - x)⁻¹` are inverses on `{x : ℝ≥0 | x < 1}`. | ||
+ `CStarAlgebra.directedOn_nonneg_ball`: the set `{x : A | 0 ≤ x} ∩ Metric.ball 0 1` is directed. | ||
## TODO | ||
+ Prove the approximate identity result by following Ken Davidson's proof in | ||
"C*-Algebras by Example" | ||
-/ | ||
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variable {A : Type*} [NonUnitalCStarAlgebra A] | ||
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local notation "σₙ" => quasispectrum | ||
local notation "σ" => spectrum | ||
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open Unitization NNReal CStarAlgebra | ||
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variable [PartialOrder A] [StarOrderedRing A] | ||
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lemma CFC.monotoneOn_one_sub_one_add_inv : | ||
MonotoneOn (cfcₙ (fun x : ℝ≥0 ↦ 1 - (1 + x)⁻¹)) (Set.Ici (0 : A)) := by | ||
intro a ha b hb hab | ||
simp only [Set.mem_Ici] at ha hb | ||
rw [← inr_le_iff .., nnreal_cfcₙ_eq_cfc_inr a _, nnreal_cfcₙ_eq_cfc_inr b _] | ||
rw [← inr_le_iff a b (.of_nonneg ha) (.of_nonneg hb)] at hab | ||
rw [← inr_nonneg_iff] at ha hb | ||
have h_cfc_one_sub (c : A⁺¹) (hc : 0 ≤ c := by cfc_tac) : | ||
cfc (fun x : ℝ≥0 ↦ 1 - (1 + x)⁻¹) c = 1 - cfc (·⁻¹ : ℝ≥0 → ℝ≥0) (1 + c) := by | ||
rw [cfc_tsub _ _ _ (fun x _ ↦ by simp) (hg := by fun_prop (disch := intro _ _; positivity)), | ||
cfc_const_one ℝ≥0 c, cfc_comp' (·⁻¹) (1 + ·) c ?_, cfc_add .., cfc_const_one ℝ≥0 c, | ||
cfc_id' ℝ≥0 c] | ||
exact continuousOn_id.inv₀ (Set.forall_mem_image.mpr fun x _ ↦ by dsimp only [id]; positivity) | ||
rw [h_cfc_one_sub (a : A⁺¹), h_cfc_one_sub (b : A⁺¹)] | ||
gcongr | ||
rw [← CFC.rpow_neg_one_eq_cfc_inv, ← CFC.rpow_neg_one_eq_cfc_inv] | ||
exact rpow_neg_one_le_rpow_neg_one (add_nonneg zero_le_one ha) (by gcongr) <| | ||
isUnit_of_le isUnit_one zero_le_one <| le_add_of_nonneg_right ha | ||
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lemma Set.InvOn.one_sub_one_add_inv : Set.InvOn (fun x ↦ 1 - (1 + x)⁻¹) (fun x ↦ x * (1 - x)⁻¹) | ||
{x : ℝ≥0 | x < 1} {x : ℝ≥0 | x < 1} := by | ||
have : (fun x : ℝ≥0 ↦ x * (1 + x)⁻¹) = fun x ↦ 1 - (1 + x)⁻¹ := by | ||
ext x : 1 | ||
field_simp | ||
simp [tsub_mul, inv_mul_cancel₀] | ||
rw [← this] | ||
constructor <;> intro x (hx : x < 1) | ||
· have : 0 < 1 - x := tsub_pos_of_lt hx | ||
field_simp [tsub_add_cancel_of_le hx.le, tsub_tsub_cancel_of_le hx.le] | ||
· field_simp [mul_tsub] | ||
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lemma norm_cfcₙ_one_sub_one_add_inv_lt_one (a : A) : | ||
‖cfcₙ (fun x : ℝ≥0 ↦ 1 - (1 + x)⁻¹) a‖ < 1 := | ||
nnnorm_cfcₙ_nnreal_lt fun x _ ↦ tsub_lt_self zero_lt_one (by positivity) | ||
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-- the calls to `fun_prop` with a discharger set off the linter | ||
set_option linter.style.multiGoal false in | ||
lemma CStarAlgebra.directedOn_nonneg_ball : | ||
DirectedOn (· ≤ ·) ({x : A | 0 ≤ x} ∩ Metric.ball 0 1) := by | ||
let f : ℝ≥0 → ℝ≥0 := fun x => 1 - (1 + x)⁻¹ | ||
let g : ℝ≥0 → ℝ≥0 := fun x => x * (1 - x)⁻¹ | ||
suffices ∀ a b : A, 0 ≤ a → 0 ≤ b → ‖a‖ < 1 → ‖b‖ < 1 → | ||
a ≤ cfcₙ f (cfcₙ g a + cfcₙ g b) by | ||
rintro a ⟨(ha₁ : 0 ≤ a), ha₂⟩ b ⟨(hb₁ : 0 ≤ b), hb₂⟩ | ||
simp only [Metric.mem_ball, dist_zero_right] at ha₂ hb₂ | ||
refine ⟨cfcₙ f (cfcₙ g a + cfcₙ g b), ⟨by simp, ?_⟩, ?_, ?_⟩ | ||
· simpa only [Metric.mem_ball, dist_zero_right] using norm_cfcₙ_one_sub_one_add_inv_lt_one _ | ||
· exact this a b ha₁ hb₁ ha₂ hb₂ | ||
· exact add_comm (cfcₙ g a) (cfcₙ g b) ▸ this b a hb₁ ha₁ hb₂ ha₂ | ||
rintro a b ha₁ - ha₂ - | ||
calc | ||
a = cfcₙ (f ∘ g) a := by | ||
conv_lhs => rw [← cfcₙ_id ℝ≥0 a] | ||
refine cfcₙ_congr (Set.InvOn.one_sub_one_add_inv.1.eqOn.symm.mono fun x hx ↦ ?_) | ||
exact lt_of_le_of_lt (le_nnnorm_of_mem_quasispectrum hx) ha₂ | ||
_ = cfcₙ f (cfcₙ g a) := by | ||
rw [cfcₙ_comp f g a ?_ (by simp [f, tsub_self]) ?_ (by simp [g]) ha₁] | ||
· fun_prop (disch := intro _ _; positivity) | ||
· have (x) (hx : x ∈ σₙ ℝ≥0 a) : 1 - x ≠ 0 := by | ||
refine tsub_pos_of_lt ?_ |>.ne' | ||
exact lt_of_le_of_lt (le_nnnorm_of_mem_quasispectrum hx) ha₂ | ||
fun_prop (disch := assumption) | ||
_ ≤ cfcₙ f (cfcₙ g a + cfcₙ g b) := by | ||
have hab' : cfcₙ g a ≤ cfcₙ g a + cfcₙ g b := le_add_of_nonneg_right cfcₙ_nonneg_of_predicate | ||
exact CFC.monotoneOn_one_sub_one_add_inv cfcₙ_nonneg_of_predicate | ||
(cfcₙ_nonneg_of_predicate.trans hab') hab' |
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