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67aee1d | 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 | import Mathlib
import OAI.Analysis.FreeWords.AmplifiedBlockModelDefinitions
import OAI.Analysis.FreeWords.AmplifiedCyclicApprox
import OAI.Analysis.FreeWords.CyclicApproxNet
import OAI.Analysis.FreeWords.MatrixPowerBlockLimits
namespace OAI
noncomputable section
namespace KKFreeLiftFrontier
open KKTraceTransfer KKMatrixExpectation KKHaarEndpoints Filter
open scoped Topology
noncomputable section
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
theorem finite_amplified_block_model : FiniteAmplifiedBlockModel (H := H) := by
classical
intro M τ hfac hdiff n hn K F E hF hE ε hε
let τn := matrixTraceMap τ.map n
let B := (numericalHom (A := ConcreteAlgebra M) n).range.toSubalgebra
obtain ⟨u,hgood,hH,hC⟩ := cyclic_approx_net τn B (FullMatrixUnitary n)
(finite_amplified_cyclic_approx M τ hfac hdiff hn)
have hblocks := matrix_power_block_limits (A := ConcreteAlgebra M)
(CyclicIndex τn B) atTop τ.map τ.normalized τ.tracial n hn u hH hC
have hKsmall : ∀ᶠj in atTop,∀k∈K,k≠0 → ‖τn (u j^k).val‖<ε := by
apply (Filter.eventually_all_finset K).2
intro k hk
by_cases hk0 : k=0
· exact Eventually.of_forall (fun _ h => (h hk0).elim)
· have hh := Metric.tendsto_nhds.mp (hH k hk0) ε hε
simpa only [dist_zero_right] using hh.mono (fun j hj _ => hj)
have hFsmall : ∀ᶠj in atTop,∀w∈F,‖τn (powerWord (u j) w)‖<ε := by
apply (Filter.eventually_all_finset F).2
intro w hw
simpa only [dist_zero_right] using Metric.tendsto_nhds.mp (hC w (hF w hw).1 (hF w hw).2) ε hε
have hEsmall : ∀ᶠj in atTop,∀z∈E,∀a b,
‖τ.map ((powerWord (u j) z.1*(u j^z.2).val) a b)‖<ε := by
apply (Filter.eventually_all_finset E).2
intro z hz
apply Filter.eventually_all.2
intro a
apply Filter.eventually_all.2
intro b
simpa only [dist_zero_right] using Metric.tendsto_nhds.mp
(hblocks z.1 (hE z hz).1 z.2 (hE z hz).2 a b) ε hε
obtain ⟨j,hjK,hjF,hjE⟩ := (hKsmall.and (hFsmall.and hEsmall)).exists
obtain ⟨d,hd,ρ,hρ,hU⟩ := hgood j
exact ⟨d,hd,ρ,hρ,u j,hU,hjK,hjF,hjE⟩
end
end KKFreeLiftFrontier
end
end OAI
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