| import OAI.Analysis.BoundedHochschild.FiniteGroupBarOperators | |
| namespace OAI | |
| /-! An explicit input formula for the bounded bar-insertion operators. -/ | |
| noncomputable section | |
| namespace BoundedHochschild.KadisonRingrose | |
| variable {M : Type*} [CStarAlgebra M] | |
| def barPrefixTuple {n : ℕ} (u : Fin (n + 2) → unitary M) (i : Fin (n + 1)) | |
| (x : Fin n → M) : Fin n → M := | |
| fun j => if j.castSucc < i then | |
| (u j.castSucc.castSucc : M) * x j * star (u j.castSucc.succ : M) else x j | |
| lemma cochainUnitaryTuple_prefix_insert {n : ℕ} (u : Fin (n + 2) → unitary M) | |
| (i : Fin (n + 1)) (x : Fin n → M) : | |
| cochainUnitaryTuple (unitaryPrefix u (i.val + 1)) (i.insertNth 1 x) = | |
| i.insertNth (u i.castSucc : M) (barPrefixTuple u i x) := by | |
| symm | |
| apply (Fin.insertNth_eq_iff).mpr | |
| constructor | |
| · simp [cochainUnitaryTuple, unitaryPrefix] | |
| · ext j | |
| change barPrefixTuple u i x j = | |
| cochainUnitaryTuple (unitaryPrefix u (i.val + 1)) (i.insertNth 1 x) (i.succAbove j) | |
| rw [show cochainUnitaryTuple (unitaryPrefix u (i.val + 1)) (i.insertNth 1 x) (i.succAbove j) = | |
| (unitaryPrefix u (i.val + 1) (i.succAbove j).castSucc : M) * x j * | |
| star (unitaryPrefix u (i.val + 1) (i.succAbove j).succ : M) by | |
| simp only [cochainUnitaryTuple, Fin.insertNth_apply_succAbove]] | |
| by_cases hji : j.castSucc < i | |
| · rw [Fin.succAbove_of_castSucc_lt i j hji] | |
| have ha : j.val < i.val + 1 := by have h := hji; simp only [Fin.lt_def, Fin.val_castSucc] at h; omega | |
| have hb : j.val + 1 < i.val + 1 := by have h := hji; simp only [Fin.lt_def, Fin.val_castSucc] at h; omega | |
| simp only [barPrefixTuple, hji, ite_true, unitaryPrefix, Fin.val_castSucc, Fin.val_succ, ha, hb] | |
| · rw [Fin.succAbove_of_le_castSucc i j (le_of_not_gt hji)] | |
| have ha : ¬j.val + 1 < i.val + 1 := by | |
| have h := hji | |
| simp only [Fin.lt_def, Fin.val_castSucc] at h | |
| omega | |
| have hb : ¬j.val + 1 + 1 < i.val + 1 := by omega | |
| simp only [barPrefixTuple, hji, ite_false, unitaryPrefix, Fin.val_castSucc, Fin.val_succ, | |
| ha, hb, OneMemClass.coe_one, star_one, one_mul, mul_one] | |
| lemma barInsertion_prefix_value {n : ℕ} (u : Fin (n + 2) → unitary M) (i : Fin (n + 1)) | |
| (f : Cochain M (n + 1)) (x : Fin n → M) : | |
| barInsertion u i f x = star (u 0 : M) * | |
| f (i.insertNth (u i.castSucc : M) (barPrefixTuple u i x)) := by | |
| rw [barInsertion_value, cochainUnitaryTuple_prefix_insert] | |
| end BoundedHochschild.KadisonRingrose | |
| end | |
| end OAI | |