| import OAI.Analysis.Cuntz.Content |
| import OAI.Analysis.Cuntz.RegularizedState |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| /-! Commutative dimension contents and continuous positive functions. -/ |
|
|
| section |
| open scoped Matrix CStarAlgebra |
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|
| namespace Kaplansky |
| attribute [local instance] Classical.propDecidable |
| attribute [local instance] CStarAlgebra.spectralOrder CStarAlgebra.spectralOrderedRing |
| variable {A X : Type*} [CStarAlgebra A] [TopologicalSpace X] |
|
|
| / |
| homomorphism. On negative functions the auxiliary value is zero and is unused. -/ |
| def CuntzDimensionState.continuousDimension (d : CuntzDimensionState A) {n : ℕ} |
| (φ : C(X, ℝ) →⋆ₐ[ℝ] CStarMatrix (Fin n) (Fin n) A) : PositiveDimensionFunctional X where |
| toFun f := if hf : 0 ≤ f then d.regularized ⟨n, CStarMatrix.ofMatrix.symm (φ f), map_nonneg φ hf⟩ else 0 |
| nonneg f hf := by simp only [dite_eq_left hf]; exact d.regularized_nonneg _ |
| mono f g hf hfg := by |
| simp only [dite_eq_left hf, dite_eq_left (hf.trans hfg)] |
| exact d.regularized_mono (cuntzLE_of_nonneg_le (map_nonneg φ hf) (OrderHomClass.mono φ hfg)) |
| subadd f g hf hg := by |
| simp only [dite_eq_left hf, dite_eq_left hg, dite_eq_left (add_nonneg hf hg), map_add] |
| exact d.regularized_sum_le _ _ (map_nonneg φ hf) (map_nonneg φ hg) |
| orthogonal f g hf hg hfg := by |
| simp only [dite_eq_left hf, dite_eq_left hg, dite_eq_left (add_nonneg hf hg), map_add] |
| apply d.regularized_orthogonal_add _ _ (map_nonneg φ hf) (map_nonneg φ hg) |
| rw [← map_mul, hfg, map_zero] |
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| @[simp] theorem CuntzDimensionState.continuousDimension_apply (d : CuntzDimensionState A) {n : ℕ} |
| (φ : C(X, ℝ) →⋆ₐ[ℝ] CStarMatrix (Fin n) (Fin n) A) (f : C(X, ℝ)) (hf : 0 ≤ f) : |
| (d.continuousDimension φ).toFun f = |
| d.regularized ⟨n, CStarMatrix.ofMatrix.symm (φ f), map_nonneg φ hf⟩ := by |
| simp [continuousDimension, hf] |
|
|
| / |
| def CuntzDimensionState.commutativeContent (d : CuntzDimensionState A) [NormalSpace X] {n : ℕ} |
| (φ : C(X, ℝ) →⋆ₐ[ℝ] CStarMatrix (Fin n) (Fin n) A) : MeasureTheory.Content X := |
| (d.continuousDimension φ).content |
|
|
| theorem CuntzDimensionState.continuousDimension_one (d : CuntzDimensionState A) {n : ℕ} |
| (φ : C(X, ℝ) →⋆ₐ[ℝ] CStarMatrix (Fin n) (Fin n) A) : |
| (d.continuousDimension φ).toFun 1 = n := by |
| rw [d.continuousDimension_apply φ 1 zero_le_one] |
| rw [← d.regularized_unit n] |
| apply d.regularized_eq |
| change CuntzEquivalent (CStarMatrix.ofMatrix.symm (φ 1)) 1 |
| rw [map_one] |
| exact .refl _ |
|
|
| end Kaplansky |
|
|
| end |
|
|
| open scoped Matrix CStarAlgebra |
|
|
| namespace Kaplansky |
| attribute [local instance] CStarAlgebra.spectralOrder CStarAlgebra.spectralOrderedRing |
| variable {A X : Type*} [CStarAlgebra A] [TopologicalSpace X] |
| namespace CuntzDimensionState |
|
|
| theorem continuousDimension_scaleInvariant (d : CuntzDimensionState A) {n : ℕ} |
| (φ : C(X, ℝ) →⋆ₐ[ℝ] CStarMatrix (Fin n) (Fin n) A) : |
| (d.continuousDimension φ).ScaleInvariant := by |
| intro f hf r hr |
| rw [d.continuousDimension_apply φ f hf, |
| d.continuousDimension_apply φ (r • f) (fun x => mul_nonneg hr.le (hf x))] |
| apply d.regularized_eq |
| change CuntzEquivalent (CStarMatrix.ofMatrix.symm (φ (r • f))) _ |
| rw [map_smul] |
| exact cuntzEquivalent_pos_smul (φ f) hr |
|
|
| theorem continuousDimension_lowerSemicontinuous (d : CuntzDimensionState A) {n : ℕ} |
| (φ : C(X, ℝ) →⋆ₐ[ℝ] CStarMatrix (Fin n) (Fin n) A) (hφ : Continuous φ) : |
| (d.continuousDimension φ).ConeLowerSemicontinuous := by |
| let g : {f : C(X, ℝ) // 0 ≤ f} → {a : CStarMatrix (Fin n) (Fin n) A // 0 ≤ a} := |
| fun f => ⟨φ f.val, map_nonneg φ f.property⟩ |
| have hg : Continuous g := (hφ.comp continuous_subtype_val).subtype_mk _ |
| have h := (d.regularized_lowerSemicontinuous n).comp hg |
| change LowerSemicontinuous (fun f : {f : C(X, ℝ) // 0 ≤ f} => (d.continuousDimension φ).toFun f.val) |
| have he : (fun f : {f : C(X, ℝ) // 0 ≤ f} => (d.continuousDimension φ).toFun f.val) = |
| (fun f : {f : C(X, ℝ) // 0 ≤ f} => d.regularized ⟨n, CStarMatrix.ofMatrix.symm (φ f.val), map_nonneg φ f.property⟩) := by |
| funext f |
| exact d.continuousDimension_apply φ f.val f.property |
| rw [he] |
| exact h |
|
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| variable [CompactSpace X] [NormalSpace X] [R1Space X] [MeasurableSpace X] [BorelSpace X] |
|
|
| / |
| Cuntz dimension of its image, with no functional or measure-existence premise. -/ |
| theorem continuousDimension_measure_positive_support (d : CuntzDimensionState A) {n : ℕ} |
| (φ : C(X, ℝ) →⋆ₐ[ℝ] CStarMatrix (Fin n) (Fin n) A) (hφ : Continuous φ) |
| (f : C(X, ℝ)) (hf : 0 ≤ f) : |
| (d.continuousDimension φ).measure {x | 0 < f x} = |
| ENNReal.ofReal (d.regularized ⟨n, CStarMatrix.ofMatrix.symm (φ f), map_nonneg φ hf⟩) := by |
| rw [← d.continuousDimension_apply φ f hf] |
| exact (d.continuousDimension φ).measure_positive_support |
| (d.continuousDimension_scaleInvariant φ) (d.continuousDimension_lowerSemicontinuous φ hφ) f hf |
|
|
| end CuntzDimensionState |
| end Kaplansky |
|
|
| end |
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| end OAI |
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