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namespace OAI
noncomputable section
open Filter
open scoped Topology CStarAlgebra ComplexOrder InnerProductSpace
namespace MainGap.Stable
variable {A : Type*} [NonUnitalCStarAlgebra A]
local instance matrixEntryOrder : PartialOrder A := CStarAlgebra.spectralOrder A
local instance matrixEntryStarOrder : StarOrderedRing A := CStarAlgebra.spectralOrderedRing A
local instance algebraicTopology' : TopologicalSpace (Algebraic A) :=
(inferInstance : PseudoMetricSpace (Algebraic A)).toUniformSpace.toTopologicalSpace
def paddedEntry (i j n : ℕ) (x : Mat A n) : A :=
if hi : i < n then if hj : j < n then x ⟨i,hi⟩ ⟨j,hj⟩ else 0 else 0
lemma paddedEntry_pad (i j : ℕ) {n m : ℕ} (h : n ≤ m) (x : Mat A n) :
paddedEntry i j m (pad h x) = paddedEntry i j n x := by
by_cases hi : i < n <;> by_cases hj : j < n
· have hi' := lt_of_lt_of_le hi h
have hj' := lt_of_lt_of_le hj h
simp [paddedEntry, hi, hj, hi', hj', pad_entry]
· by_cases hi' : i < m <;> by_cases hj' : j < m <;>
simp [paddedEntry, hi, hj, hi', hj', pad_entry]
· by_cases hi' : i < m <;> by_cases hj' : j < m <;>
simp [paddedEntry, hi, hi', hj', pad_entry]
· by_cases hi' : i < m <;> by_cases hj' : j < m <;>
simp [paddedEntry, hi, hi', hj', pad_entry]
def entryAlgLinear (i j : ℕ) : Algebraic A →ₗ[ℂ] A where
toFun := DirectLimit.lift _ (paddedEntry i j) (fun _ _ h x => (paddedEntry_pad i j h x).symm)
map_add' := DirectLimit.induction₂ _ fun n x y => by
rw [DirectLimit.add_def]
change paddedEntry i j n (x+y) = paddedEntry i j n x + paddedEntry i j n y
by_cases hi : i < n <;> by_cases hj : j < n <;> simp [paddedEntry, hi, hj]
map_smul' c := DirectLimit.induction _ fun n x => by
rw [DirectLimit.smul_def]
change paddedEntry i j n (c • x) = c • paddedEntry i j n x
by_cases hi : i < n <;> by_cases hj : j < n <;> simp [paddedEntry, hi, hj]
@[simp] lemma entryAlgLinear_corner (i j n : ℕ) (x : Mat A n) :
entryAlgLinear i j (cornerAlg n x) = paddedEntry i j n x := rfl
lemma norm_entryAlgLinear_le (i j : ℕ) (x : Algebraic A) :
‖entryAlgLinear i j x‖ ≤ ‖x‖ := by
induction x using DirectLimit.induction with
| ih n x =>
change ‖paddedEntry i j n x‖ ≤ ‖x‖
by_cases hi : i < n <;> by_cases hj : j < n
· simpa [paddedEntry, hi, hj] using
(CStarMatrix.norm_entry_le_norm (M := x) (i := ⟨i,hi⟩) (j := ⟨j,hj⟩))
all_goals simp [paddedEntry, hi, hj]
def entryAlg (i j : ℕ) : Algebraic A →L[ℂ] A :=
(entryAlgLinear i j).mkContinuous 1 (fun x => by simpa using norm_entryAlgLinear_le i j x)
def entry (i j : ℕ) : Stabilization A → A :=
UniformSpace.Completion.extension (entryAlg (A := A) i j)
lemma continuous_entry (i j : ℕ) : Continuous (entry (A := A) i j) :=
UniformSpace.Completion.continuous_extension
@[simp] lemma entry_coe (i j : ℕ) (x : Algebraic A) :
entry i j (x : Stabilization A) = entryAlg (A := A) i j x := by
unfold entry
exact UniformSpace.Completion.extension_coe (f := (entryAlg (A := A) i j))
(ContinuousLinearMap.uniformContinuous (entryAlg (A := A) i j)) x
@[simp] lemma entry_corner (i j n : ℕ) (x : Mat A n) :
entry i j (corner n x) = paddedEntry i j n x := by
rw [corner_apply, entry_coe]
rfl
def block (m n : ℕ) (x : Stabilization A) : CStarMatrix (Fin m) (Fin n) A :=
CStarMatrix.ofMatrix fun i j => entry i.val j.val x
@[simp] lemma block_apply (m n : ℕ) (x : Stabilization A) (i : Fin m) (j : Fin n) :
block m n x i j = entry i.val j.val x := rfl
lemma continuous_block (m n : ℕ) : Continuous (block (A := A) m n) := by
exact continuous_pi fun i => continuous_pi fun j => (continuous_entry i.val j.val)
@[simp] lemma block_corner_self (n : ℕ) (x : Mat A n) : block n n (corner n x) = x := by
ext i j
simp only [block_apply, entry_corner, paddedEntry, dite_eq_left i.isLt, dite_eq_left j.isLt]
end MainGap.Stable
namespace MainGap.Stable
variable {A : Type*} [NonUnitalCStarAlgebra A]
local instance blockConjugateOrder : PartialOrder A := CStarAlgebra.spectralOrder A
local instance blockConjugateStarOrder : StarOrderedRing A := CStarAlgebra.spectralOrderedRing A
local instance algebraicTopology'' : TopologicalSpace (Algebraic A) :=
(inferInstance : PseudoMetricSpace (Algebraic A)).toUniformSpace.toTopologicalSpace
lemma paddedEntry_star (i j n : ℕ) (x : Mat A n) :
paddedEntry i j n (star x) = star (paddedEntry j i n x) := by
by_cases hi : i < n <;> by_cases hj : j < n <;>
simp [paddedEntry, hi, hj, CStarMatrix.star_apply]
lemma paddedEntry_mul (i j n : ℕ) (x y : Mat A n) :
paddedEntry i j n (x*y) =
∑ k : Fin n, paddedEntry i k.val n x * paddedEntry k.val j n y := by
by_cases hi : i < n <;> by_cases hj : j < n <;>
simp [paddedEntry, hi, hj, CStarMatrix.mul_apply]
lemma paddedEntry_conjugate (i j : ℕ) {m N : ℕ} (h : m ≤ N)
(z : Mat A N) (v : Mat A m) :
paddedEntry i j N (star z * pad h v * z) =
∑ q : Fin m, ∑ p : Fin m,
star (paddedEntry p.val i N z) * v p q * paddedEntry q.val j N z := by
classical
simp only [paddedEntry_mul, paddedEntry_star, paddedEntry_pad]
rw [← sum_pad h (fun q : Fin m => ∑ p : Fin m,
star (paddedEntry p.val i N z) * v p q * paddedEntry q.val j N z)]
apply Finset.sum_congr rfl
intro q _
by_cases hq : q.val < m
· rw [dite_eq_left hq, Finset.sum_mul]
rw [← sum_pad h (fun p : Fin m =>
star (paddedEntry p.val i N z) * v p ⟨q,hq⟩ * paddedEntry q.val j N z)]
apply Finset.sum_congr rfl
intro p _
by_cases hp : p.val < m <;> simp [paddedEntry, hp, hq]
· simp [paddedEntry, hq]
lemma block_conjugate_corner {m N : ℕ} (h : m ≤ N) (n : ℕ)
(v : Mat A m) (z : Mat A N) :
block n n (star (corner N z) * corner m v * corner N z) =
(block m n (corner N z)).conjTranspose * v * block m n (corner N z) := by
rw [← corner_pad h v]
have he : star (corner N z) * corner N (pad h v) * corner N z =
corner N (star z * pad h v * z) := by rw [map_mul, map_mul, map_star]
rw [he]
ext i j
simp only [block_apply, entry_corner, CStarMatrix.mul_apply,
CStarMatrix.conjTranspose_apply, Finset.sum_mul]
exact paddedEntry_conjugate i.val j.val h z v
lemma block_conjugate (m n : ℕ) (v : Mat A m) (x : Stabilization A) :
block n n (star x * corner m v * x) =
(block m n x).conjTranspose * v * block m n x := by
refine UniformSpace.Completion.induction_on x ?_ ?_
· apply isClosed_eq
· exact (continuous_block n n).comp ((continuous_star.mul continuous_const).mul continuous_id)
· have hc : Continuous (fun x : Stabilization A => (block m n x).conjTranspose) := by
change Continuous (fun x : Stabilization A =>
fun i j => star (block m n x j i))
exact continuous_pi fun i => continuous_pi fun j =>
continuous_star.comp (continuous_entry j.val i.val)
exact (hc.matrix_mul continuous_const).matrix_mul (continuous_block m n)
· intro z
induction z using DirectLimit.induction with
| ih k z =>
change block n n (star (corner k z) * corner m v * corner k z) =
(block m n (corner k z)).conjTranspose * v * block m n (corner k z)
rw [← corner_pad (le_max_right m k) z]
exact block_conjugate_corner (le_max_left m k) n v _
end MainGap.Stable
end
end OAI
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