| 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 | |