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a458723 | 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 | import Mathlib
import OAI.Analysis.Commutator.TracialConjugation
import OAI.Analysis.Commutator.TracialL2Tools
namespace OAI
noncomputable section
section
noncomputable section
open scoped ComplexOrder
namespace Kadison.TracialL2
open TracialGNS
variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A]
variable (τ : A →ₚ[ℂ] ℂ)
lemma exists_centered_contraction_dense_sequence [TopologicalSpace.SeparableSpace τ.GNS] :
∃ x : ℕ → A, (∀ j,‖x j‖ ≤ 1) ∧ (∀ j,τ (x j)=0) ∧
∀ a : A,‖a‖ ≤ 1 → τ a=0 → ∀ ε : ℝ,0<ε → ∃ j,norm τ (a-x j)<ε := by
classical
let C := {a : A // ‖a‖ ≤ 1 ∧ τ a=0}
let S : Set τ.GNS := Set.range (fun a : C => algebraVector τ a.val)
have : Nonempty S :=
⟨⟨algebraVector τ 0,⟨⟨0,by simp⟩,rfl⟩⟩⟩
obtain ⟨u,hu⟩ := TopologicalSpace.exists_dense_seq S
choose x hx using (fun n => (u n).property)
refine ⟨fun n => (x n).val,fun n => (x n).property.1,fun n => (x n).property.2,?_⟩
intro a ha hc ε hε
obtain ⟨j,hj⟩ := hu.exists_dist_lt ⟨algebraVector τ a,⟨⟨a,ha,hc⟩,rfl⟩⟩ hε
refine ⟨j,?_⟩
change dist (algebraVector τ a) (u j).val<ε at hj
rw [←hx j,dist_eq_norm,←map_sub] at hj
simpa only [algebraVector_apply,UniformSpace.Completion.norm_coe,norm] using hj
end Kadison.TracialL2
end
end
end
end OAI
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