| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| section | |
| noncomputable section | |
| open scoped ComplexOrder ContinuousFunctionalCalculus | |
| namespace Kadison.CFCIntertwiner | |
| variable {A : Type*} [CStarAlgebra A] | |
| local instance : ContinuousFunctionalCalculus ℂ (A × A) IsStarNormal := | |
| IsStarNormal.instContinuousFunctionalCalculus | |
| local instance : ContinuousFunctionalCalculus ℝ (A × A) IsSelfAdjoint := | |
| IsSelfAdjoint.instContinuousFunctionalCalculus | |
| lemma semiconj_cfc (x a b : A) (ha : IsSelfAdjoint a) (hb : IsSelfAdjoint b) | |
| (h : x*a=b*x) (f : ℝ → ℝ) (hf : Continuous f) : | |
| x*cfc f a=cfc f b*x := by | |
| have hab : IsSelfAdjoint (a,b) := Prod.ext ha hb | |
| have hc : ∀ g : C(spectrum ℝ (a,b),ℝ), | |
| x*(cfcHom hab g).1=(cfcHom hab g).2*x := by | |
| intro g | |
| induction g using ContinuousMap.induction_on_of_compact with | |
| | const r => | |
| change x*((cfcHom hab) (algebraMap ℝ _ r)).1=((cfcHom hab) (algebraMap ℝ _ r)).2*x | |
| rw [AlgHomClass.commutes] | |
| change x*algebraMap ℝ A r=algebraMap ℝ A r*x | |
| exact (Algebra.commutes r x).symm | |
| | id => simpa only [cfcHom_id] using h | |
| | star_id => simpa only [map_star,cfcHom_id,Prod.fst_star,Prod.snd_star,ha.star_eq,hb.star_eq] using h | |
| | add g k hg hk => simp only [map_add,Prod.fst_add,Prod.snd_add,mul_add,add_mul,hg,hk] | |
| | mul g k hg hk => | |
| simp only [map_mul,Prod.fst_mul,Prod.snd_mul] | |
| calc | |
| x*((cfcHom hab g).1*(cfcHom hab k).1)= | |
| (x*(cfcHom hab g).1)*(cfcHom hab k).1 := (mul_assoc ..).symm | |
| _=((cfcHom hab g).2*x)*(cfcHom hab k).1 := by rw [hg] | |
| _=(cfcHom hab g).2*((cfcHom hab k).2*x) := by rw [mul_assoc,hk] | |
| _=((cfcHom hab g).2*(cfcHom hab k).2)*x := (mul_assoc ..).symm | |
| | frequently g hg => | |
| have closed : IsClosed {k : C(spectrum ℝ (a,b),ℝ) | | |
| x*(cfcHom hab k).1=(cfcHom hab k).2*x} := | |
| isClosed_eq ((cfcHom_continuous hab).fst.const_mul x) | |
| ((cfcHom_continuous hab).snd.mul_const x) | |
| exact closed.mem_of_frequently_of_tendsto hg Filter.tendsto_id | |
| have hh : x*(cfc f (a,b)).1=(cfc f (a,b)).2*x := by | |
| rw [cfc_apply f (a,b) hab hf.continuousOn] | |
| exact hc _ | |
| rw [cfc_map_prod (S := ℂ) f a b hf.continuousOn hab ha hb] at hh | |
| exact hh | |
| end Kadison.CFCIntertwiner | |
| end | |
| end | |
| end | |
| end OAI | |