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9eeb9c7 | 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 | import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.LocallyConvex.SeparatingDual
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
namespace OAI
namespace Invariant233
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H]
theorem centralizer_singleton_eq_top_iff_scalar (T : H →L[ℂ] H) :
Subalgebra.centralizer ℂ ({T} : Set (H →L[ℂ] H)) = ⊤ ↔
∃ c : ℂ, T = c • (1 : H →L[ℂ] H) := by
rw [Subalgebra.centralizer_eq_top_iff_subset]
simp only [Set.singleton_subset_iff, Algebra.IsCentral.center_eq_bot]
simp [Algebra.algebraMap_eq_smul_one, eq_comm]
theorem centralizer_singleton_ne_top_of_nonscalar (T : H →L[ℂ] H)
(hT : ¬ ∃ c : ℂ, T = c • (1 : H →L[ℂ] H)) :
Subalgebra.centralizer ℂ ({T} : Set (H →L[ℂ] H)) ≠ ⊤ := by
exact fun h => hT ((centralizer_singleton_eq_top_iff_scalar T).mp h)
variable {A : Type*} [Ring A] [Algebra ℂ A]
theorem centralizer_singleton_add_scalar (T : A) (c : ℂ) :
Subalgebra.centralizer ℂ ({T + c • (1 : A)} : Set A) =
Subalgebra.centralizer ℂ ({T} : Set A) := by
ext S
simp only [Subalgebra.mem_centralizer_iff, Set.mem_singleton_iff,
forall_eq, add_mul, mul_add, smul_mul_assoc, mul_smul_comm,
one_mul, mul_one, add_right_cancel_iff]
theorem centralizer_singleton_one_add (T : A) :
Subalgebra.centralizer ℂ ({1 + T} : Set A) =
Subalgebra.centralizer ℂ ({T} : Set A) := by
simpa only [one_smul, add_comm] using centralizer_singleton_add_scalar T 1
end Invariant233
end OAI
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