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f1f7953 | 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 | 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
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