| import Mathlib | |
| import OAI.Analysis.TracialSplitting.PreparedBlocks | |
| import OAI.Analysis.JiangSu.Model | |
| namespace OAI | |
| noncomputable section | |
| open scoped NonUnitalContinuousFunctionalCalculus Topology | |
| open ContinuousMapZero Filter | |
| namespace TraceCone | |
| variable {A : Type*} [NonUnitalCStarAlgebra A] [PartialOrder A] [StarOrderedRing A] | |
| theorem cfc_intertwine (a b : Positive A) (x : A) (hx : x * a.1 = b.1 * x) | |
| (f : ℝ → ℝ) (hf : Continuous f) (hf0 : f 0 = 0) : | |
| x * cfcₙ f a.1 = cfcₙ f b.1 * x := by | |
| let s := quasispectrum ℝ a.1 ∪ quasispectrum ℝ b.1 | |
| have : CompactSpace s := isCompact_iff_compactSpace.mp | |
| ((NonUnitalContinuousFunctionalCalculus.isCompact_quasispectrum a.1).union (NonUnitalContinuousFunctionalCalculus.isCompact_quasispectrum b.1)) | |
| have : Fact (0 ∈ s) := ⟨Or.inl (quasispectrum.zero_mem ℝ a.1)⟩ | |
| have has : quasispectrum ℝ a.1 ⊆ s := Set.subset_union_left | |
| have hbs : quasispectrum ℝ b.1 ⊆ s := Set.subset_union_right | |
| have ha := IsSelfAdjoint.of_nonneg a.2 | |
| have hb := IsSelfAdjoint.of_nonneg b.2 | |
| let F := cfcₙHomSuperset ha has | |
| let G := cfcₙHomSuperset hb hbs | |
| have hF : Continuous F := cfcₙHomSuperset_continuous ha has | |
| have hG : Continuous G := cfcₙHomSuperset_continuous hb hbs | |
| have key (g : C(s, ℝ)₀) : x * F g = G g * x := by | |
| induction g using ContinuousMapZero.induction_on_of_compact with | |
| | zero => simp | |
| | smul r g hg => | |
| rw [map_smul, map_smul, mul_smul_comm, smul_mul_assoc, hg] | |
| | id => simpa only [F, G, cfcₙHomSuperset_id] using hx | |
| | star_id => simpa only [star_trivial, F, G, cfcₙHomSuperset_id] using hx | |
| | add g h hg hh => simp only [map_add, mul_add, add_mul, hg, hh] | |
| | mul g h hg hh => | |
| rw [map_mul, map_mul, ← mul_assoc, hg, mul_assoc, hh, ← mul_assoc] | |
| | frequently g hg => | |
| have hc : IsClosed {g : C(s, ℝ)₀ | x * F g = G g * x} := | |
| isClosed_eq (continuous_const.mul hF) (hG.mul continuous_const) | |
| exact hc.mem_of_frequently_of_tendsto hg tendsto_id | |
| let g : C(s, ℝ)₀ := ⟨⟨fun t => f t.1, hf.comp continuous_subtype_val⟩, by change f 0 = 0; exact hf0⟩ | |
| have h := key g | |
| have heq (c : Positive A) (hc : quasispectrum ℝ c.1 ⊆ s) : | |
| cfcₙHomSuperset (IsSelfAdjoint.of_nonneg c.2) hc g = cfcₙ f c.1 := by | |
| rw [cfcₙHomSuperset_apply, cfcₙHom_eq_cfcₙ_extend f] | |
| apply cfcₙ_congr | |
| intro t ht | |
| lift t to quasispectrum ℝ c.1 using ht | |
| simp only [Subtype.val_injective.extend_apply, comp_apply, coe_mk, | |
| ContinuousMap.coe_mk, g] | |
| rfl | |
| change x * cfcₙHomSuperset ha has g = cfcₙHomSuperset hb hbs g * x at h | |
| simpa only [heq a has, heq b hbs] using h | |
| end TraceCone | |
| end | |
| end OAI | |