openAI-math-dataset / lean /OAI /Analysis /JiangSu /DimensionFunctions /FiniteSpectralIntegralOrthogonal.lean
| import Mathlib | |
| import OAI.Analysis.TracialSplitting.PreparedBlocks | |
| import OAI.Analysis.JiangSu.DimensionFunctions.FiniteSpectralIntegral | |
| import OAI.Analysis.JiangSu.CuntzSemigroup.FiniteCuntzSums | |
| namespace OAI | |
| noncomputable section | |
| namespace Pureness | |
| open TraceCone MeasureTheory Set | |
| open scoped ENNReal NNReal CStarAlgebra | |
| variable {D : Type*} [NonUnitalCStarAlgebra D] [PartialOrder D] [StarOrderedRing D] | |
| lemma positive_mul_power_zero (a b : Positive D) (hab : a.1*b.1 = 0) | |
| (r : ℝ≥0) (hr : 0 < r) : a.1 * (Positive.power r b).1 = 0 := by | |
| have hf : Continuous (fun t : ℝ => t^(r : ℝ)) := continuous_id.rpow_const (fun _ => Or.inr r.2) | |
| have hz : (0 : ℝ)^(r : ℝ) = 0 := Real.zero_rpow (by exact_mod_cast hr.ne') | |
| have h := cfc_intertwine b Positive.zero a.1 (by simpa [Positive.zero] using hab) | |
| (fun t : ℝ => t^(r : ℝ)) hf hz | |
| change a.1 * b.1 ^ r = 0 | |
| rw [CFC.nnrpow_eq_cfcₙ_real b.1 r b.2] | |
| simpa [Positive.zero] using h | |
| lemma positive_sqrt_mul_zero (a b : Positive D) (hab : a.1*b.1 = 0) : | |
| CFC.sqrt a.1 * CFC.sqrt b.1 = 0 := by | |
| have h := positive_mul_power_zero a b hab (1/2) (by norm_num) | |
| have hs : (Positive.power (1/2) b).1*a.1 = 0 := by | |
| simpa only [star_mul,a.2.star_eq,(Positive.power (1/2) b).2.star_eq,star_zero] using congrArg star h | |
| have ht := positive_mul_power_zero (Positive.power (1/2) b) a hs (1/2) (by norm_num) | |
| have hu := congrArg star ht | |
| have hv : (Positive.power (1/2) a).1 * (Positive.power (1/2) b).1 = 0 := by | |
| simpa only [star_mul,(Positive.power (1/2) a).2.star_eq, | |
| (Positive.power (1/2) b).2.star_eq,star_zero] using hu | |
| simpa only [Positive.power,CFC.nnrpow_eq_pow,← CFC.sqrt_eq_nnrpow] using hv | |
| private noncomputable abbrev positiveSqrt (a : Positive D) : D := CFC.sqrt a.1 | |
| private theorem positiveSqrt_star (a : Positive D) : star (positiveSqrt a) = positiveSqrt a := | |
| (CFC.sqrt_nonneg a.1).star_eq | |
| private theorem positiveSqrt_mul (a : Positive D) : positiveSqrt a * positiveSqrt a = a.1 := | |
| CFC.sqrt_mul_sqrt_self a.1 a.2 | |
| variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] | |
| def finiteOrthogonalColumn (n : ℕ) (M N : FinMatrix A n) : | |
| CStarMatrix (Fin n × Fin 2) (Fin n × Fin 2) A := | |
| fun i j => if j.2 = 0 then if i.2 = 0 then M i.1 j.1 else N i.1 j.1 else 0 | |
| omit [PartialOrder A] [StarOrderedRing A] in | |
| lemma finiteOrthogonalColumn_starMul (n : ℕ) (M N : FinMatrix A n) : | |
| star (finiteOrthogonalColumn n M N)*finiteOrthogonalColumn n M N = | |
| matrixInterleaveRaw n (star M*M+star N*N,0) := by | |
| apply CStarMatrix.ext | |
| rintro ⟨i,r⟩ ⟨j,s⟩ | |
| fin_cases r <;> fin_cases s <;> | |
| simp [finiteOrthogonalColumn,matrixInterleaveRaw_entry',CStarMatrix.mul_apply,CStarMatrix.star_apply, | |
| Fintype.sum_prod_type,Fin.sum_univ_two,CStarMatrix.add_apply,Finset.sum_add_distrib,apply_ite star] | |
| omit [PartialOrder A] [StarOrderedRing A] in | |
| lemma finiteOrthogonalColumn_mulStar (n : ℕ) (M N : FinMatrix A n) | |
| (hMN : M*star N = 0) (hNM : N*star M = 0) : | |
| finiteOrthogonalColumn n M N*star (finiteOrthogonalColumn n M N) = | |
| matrixInterleaveRaw n (M*star M,N*star N) := by | |
| apply CStarMatrix.ext | |
| rintro ⟨i,r⟩ ⟨j,s⟩ | |
| have h1 := congrArg (fun P : FinMatrix A n => P i j) hMN | |
| have h2 := congrArg (fun P : FinMatrix A n => P i j) hNM | |
| fin_cases r <;> fin_cases s <;> | |
| simp_all [finiteOrthogonalColumn,matrixInterleaveRaw_entry',CStarMatrix.mul_apply,CStarMatrix.star_apply, | |
| Fintype.sum_prod_type,apply_ite star] | |
| theorem finiteSpectralIntegral_add_orthogonal | |
| (d : finiteCuntzSubmonoid (A := A) →+ ℝ≥0∞) (hd : Monotone d) | |
| (n : ℕ) (M N : Positive (FinMatrix A n)) (hMN : M.1*N.1 = 0) : | |
| finiteSpectralIntegral d n (Positive.add M N) = | |
| finiteSpectralIntegral d n M + finiteSpectralIntegral d n N := by | |
| let e := CStarMatrix.reindexₐ ℂ A (finProdFinEquiv : (Fin n × Fin 2) ≃ Fin (n*2)) | |
| let y := e (finiteOrthogonalColumn n (positiveSqrt M) (positiveSqrt N)) | |
| have hcross := positive_sqrt_mul_zero M N hMN | |
| have hcross' : positiveSqrt N * positiveSqrt M = 0 := by | |
| simpa only [star_mul,positiveSqrt_star M,positiveSqrt_star N,star_zero] using congrArg star hcross | |
| have hl : Positive.starMul y = Positive.map (matrixInterleave n) (positiveProduct (Positive.add M N) Positive.zero) := by | |
| apply Subtype.ext | |
| change star (e (finiteOrthogonalColumn n (positiveSqrt M) (positiveSqrt N))) * | |
| e (finiteOrthogonalColumn n (positiveSqrt M) (positiveSqrt N)) = | |
| e (matrixInterleaveRaw n (M.1+N.1,0)) | |
| rw [← map_star,← map_mul,finiteOrthogonalColumn_starMul, | |
| positiveSqrt_star M,positiveSqrt_star N, | |
| positiveSqrt_mul M,positiveSqrt_mul N] | |
| have hr : Positive.mulStar y = Positive.map (matrixInterleave n) (positiveProduct M N) := by | |
| apply Subtype.ext | |
| change e (finiteOrthogonalColumn n (positiveSqrt M) (positiveSqrt N)) * | |
| star (e (finiteOrthogonalColumn n (positiveSqrt M) (positiveSqrt N))) = | |
| e (matrixInterleaveRaw n (M.1,N.1)) | |
| rw [← map_star,← map_mul,finiteOrthogonalColumn_mulStar n _ _ | |
| (by simpa only [positiveSqrt_star N] using hcross) | |
| (by simpa only [positiveSqrt_star M] using hcross'), | |
| positiveSqrt_star M,positiveSqrt_star N, | |
| positiveSqrt_mul M,positiveSqrt_mul N] | |
| have h := finiteSpectralIntegral_polar d (n*2) y | |
| rw [hl,hr,finiteSpectralIntegral_block_add d hd,finiteSpectralIntegral_block_add d hd, | |
| finiteSpectralIntegral_zero,add_zero] at h | |
| exact h | |
| end Pureness | |
| end | |
| end OAI | |