| import Mathlib.Analysis.VonNeumannAlgebra.Basic |
| import Mathlib.Analysis.InnerProductSpace.Adjoint |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace FiniteFactor |
| open scoped Topology |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
|
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| / |
| representation. These are operator-algebra data, without approximation assumptions. -/ |
| structure TracialRepresentation (M : VonNeumannAlgebra H) where |
| vector : H |
| norm_vector : ‖vector‖ = 1 |
| cyclic : ∀ S T : M.toStarSubalgebra, |
| inner ℂ vector ((S : H →L[ℂ] H) ((T : H →L[ℂ] H) vector)) = |
| inner ℂ vector ((T : H →L[ℂ] H) ((S : H →L[ℂ] H) vector)) |
| separating : ∀ T : M.toStarSubalgebra, (T : H →L[ℂ] H) vector = 0 → T = 0 |
|
|
| namespace TracialRepresentation |
| variable {M : VonNeumannAlgebra H} (τ : TracialRepresentation M) |
| abbrev Algebra := M.toStarSubalgebra |
|
|
| def trace (T : Algebra (M := M)) : ℂ := inner ℂ τ.vector ((T : H →L[ℂ] H) τ.vector) |
| def norm₂ (T : Algebra (M := M)) : ℝ := ‖(T : H →L[ℂ] H) τ.vector‖ |
|
|
| @[simp] theorem trace_one : τ.trace 1 = 1 := by |
| simp [trace, inner_self_eq_norm_sq_to_K, τ.norm_vector] |
|
|
| theorem trace_mul_comm (S T : Algebra (M := M)) : τ.trace (S*T) = τ.trace (T*S) := |
| τ.cyclic S T |
|
|
| theorem trace_star_mul (T : Algebra (M := M)) : |
| τ.trace (star T*T) = (τ.norm₂ T : ℂ)^2 := by |
| unfold trace norm₂ |
| change inner ℂ τ.vector ((star (T : H →L[ℂ] H)) ((T : H →L[ℂ] H) τ.vector)) = _ |
| rw [ContinuousLinearMap.star_eq_adjoint, ContinuousLinearMap.adjoint_inner_right, inner_self_eq_norm_sq_to_K] |
| norm_cast |
|
|
| theorem norm₂_star (T : Algebra (M := M)) : τ.norm₂ (star T) = τ.norm₂ T := by |
| have h := τ.trace_mul_comm (star T) T |
| have h2 := τ.trace_star_mul (star T) |
| simp only [star_star] at h2 |
| rw [τ.trace_star_mul, h2] at h |
| have hr := congrArg Complex.re h |
| simp only [← Complex.ofReal_pow, Complex.ofReal_re] at hr |
| have hn : 0 ≤ τ.norm₂ T := norm_nonneg _ |
| have hs : 0 ≤ τ.norm₂ (star T) := norm_nonneg _ |
| nlinarith |
|
|
| @[simp] theorem norm₂_zero : τ.norm₂ 0 = 0 := by simp [norm₂] |
| @[simp] theorem norm₂_sub (S T : Algebra (M := M)) : |
| τ.norm₂ (S-T) = ‖(S : H →L[ℂ] H) τ.vector - (T : H →L[ℂ] H) τ.vector‖ := rfl |
|
|
| theorem norm₂_add_le (S T : Algebra (M := M)) : τ.norm₂ (S+T) ≤ τ.norm₂ S+τ.norm₂ T := |
| norm_add_le _ _ |
|
|
| theorem norm₂_smul (c : ℂ) (T : Algebra (M := M)) : τ.norm₂ (c • T) = ‖c‖*τ.norm₂ T := |
| norm_smul _ _ |
|
|
| theorem norm₂_mul_le (S T : Algebra (M := M)) : τ.norm₂ (S*T) ≤ ‖S‖ * τ.norm₂ T := |
| (S : H →L[ℂ] H).le_opNorm ((T : H →L[ℂ] H) τ.vector) |
|
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| theorem norm₂_mul_le_right (S T : Algebra (M := M)) : τ.norm₂ (S*T) ≤ τ.norm₂ S * ‖T‖ := by |
| rw [← τ.norm₂_star (S*T), star_mul] |
| have h := τ.norm₂_mul_le (star T) (star S) |
| simpa [τ.norm₂_star, mul_comm, ContinuousLinearMap.star_eq_adjoint] using h |
|
|
| theorem norm₂_le_norm (T : Algebra (M := M)) : τ.norm₂ T ≤ ‖T‖ := by |
| have h := (T : H →L[ℂ] H).le_opNorm τ.vector |
| simpa [norm₂, τ.norm_vector] using h |
|
|
| theorem norm_trace_le_norm₂ (T : Algebra (M := M)) : ‖τ.trace T‖ ≤ τ.norm₂ T := by |
| simpa [trace, norm₂, τ.norm_vector] using norm_inner_le_norm τ.vector ((T : H →L[ℂ] H) τ.vector) |
|
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| theorem norm₂_eq_zero_iff (T : Algebra (M := M)) : τ.norm₂ T = 0 ↔ T = 0 := by |
| constructor |
| · intro h |
| exact τ.separating T (norm_eq_zero.mp h) |
| · rintro rfl |
| exact τ.norm₂_zero |
|
|
| end TracialRepresentation |
|
|
| end FiniteFactor |
|
|
| end |
|
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| end OAI |
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|
