| 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 |
|
|
