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9eeb9c7 | 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 | import OAI.Analysis.IrrationalRotation.Model.NormalAffiliation
namespace OAI
noncomputable section
open Filter
open scoped Topology
namespace Invariant233
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
[CompleteSpace E]
theorem IsFaithfulTracialVector.isNormalFaithfulTracialState
{M : VonNeumannAlgebra E} {ξ : E} (hξ : IsFaithfulTracialVector M ξ) :
IsNormalFaithfulTracialState M hξ.toTracialState := by
constructor
· intro a ha hzero
apply Subtype.ext
exact hξ.2.2 (a : E →L[ℂ] E) a.property ha hzero
· intro ι L _hL a b _hbound hstrong
change Tendsto (fun i => inner ℂ ξ ((a i : E →L[ℂ] E) ξ)) L
(𝓝 (inner ℂ ξ ((b : E →L[ℂ] E) ξ)))
exact tendsto_const_nhds.inner (hstrong ξ)
end Invariant233
end
end OAI
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