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9655094 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 | import OAI.Analysis.CharacterCriterion.Definitions
import OAI.Analysis.CharacterCriterion.JiangSu
namespace OAI
noncomputable section
namespace KirchbergRordam
lemma nonzero_character_one {A : Type*} [CStarAlgebra A]
(χ : A →⋆ₙₐ[ℂ] ℂ) (hχ : χ≠0) : χ 1=1 := by
have hn : χ 1≠0 := by
intro h
apply hχ
ext a
change χ a=0
calc
χ a = χ (1*a) := by rw [one_mul]
_ = χ 1*χ a := map_mul χ _ _
_ = 0 := by rw [h,zero_mul]
apply mul_left_cancel₀ hn
rw [←map_mul χ,one_mul,mul_one]
def unitalCharacter {A : Type*} [CStarAlgebra A]
(χ : A →⋆ₙₐ[ℂ] ℂ) (hχ : χ≠0) : A →⋆ₐ[ℂ] ℂ where
toFun := χ
map_zero' := map_zero χ
map_one' := nonzero_character_one χ hχ
map_add' := map_add χ
map_mul' := map_mul χ
commutes' z := by
change χ (algebraMap ℂ A z)=z
rw [Algebra.algebraMap_eq_smul_one,map_smul,nonzero_character_one χ hχ,
smul_eq_mul,mul_one]
map_star' := map_star χ
lemma hasNoCharacters_iff (A : Type*) [CStarAlgebra A] :
HasNoCharacters A ↔ IsEmpty (A →⋆ₐ[ℂ] ℂ) := by
classical
constructor
· intro h
refine ⟨fun χ => ?_⟩
have he := DFunLike.congr_fun (h χ.toNonUnitalStarAlgHom) 1
change χ 1=0 at he
rw [map_one] at he
exact one_ne_zero he
· intro h χ
by_contra hn
exact h.false (unitalCharacter χ hn)
attribute [local instance 1100] StarCompletion.completionStar
theorem character_criterion
(A : Type) [CStarAlgebra A] [Nontrivial A] [TopologicalSpace.SeparableSpace A]
(ω : Ultrafilter ℕ) (hω : (ω : Filter ℕ)≤Filter.cofinite) :
HasNoCharacters (NormUltrapower.CentralAlgebra A ω) ↔
Nonempty (A ≃⋆ₐ[ℂ] MinTensor.Algebra A JiangSu.Algebra) := by
rw [hasNoCharacters_iff]
exact StandardPrimeModel.presentation.character_criterion A ω hω
end KirchbergRordam
end
end OAI
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