| import OAI.Analysis.GroupDeterminants.RegularAdjoint |
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| namespace OAI |
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| universe uA |
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| noncomputable section |
| open scoped BigOperators |
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| namespace GroupRingDeterminant |
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| section ProjectionLog |
| variable {A : Type uA} [CStarAlgebra A] |
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| theorem cfc_projectionReweight (P : A) (hP : IsSelfAdjoint P) (a b : ℝ) : |
| cfc (fun x : ℝ => a * x + b * (1 - x)) P = projectionReweight P a b := by |
| rw [cfc_add P (fun x : ℝ => a * x) (fun x : ℝ => b * (1 - x)) |
| (by fun_prop) (by fun_prop), |
| cfc_const_mul a (fun x : ℝ => x) P (by fun_prop), |
| cfc_const_mul b (fun x : ℝ => 1 - x) P (by fun_prop), |
| cfc_sub (fun _ : ℝ => 1) (fun x : ℝ => x) P (by fun_prop) (by fun_prop)] |
| simp [projectionReweight, cfc_id' ℝ P hP, cfc_const_one ℝ P hP] |
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| theorem log_projectionReweight (P : A) (hP : IsSelfAdjoint P) (hPP : P * P = P) |
| (a b : ℝ) (ha : a ≠ 0) (hb : b ≠ 0) : |
| CFC.log (projectionReweight P a b) = |
| projectionReweight P (Real.log a) (Real.log b) := by |
| have hp : IsIdempotentElem P := hPP |
| have hc : ContinuousOn Real.log ((fun x : ℝ => a * x + b * (1 - x)) '' spectrum ℝ P) := by |
| rintro y ⟨x, hx, rfl⟩ |
| rcases hp.spectrum_subset ℝ hx with h | h |
| · have hx0 : x = 0 := h |
| simp only [hx0, mul_zero, sub_zero, mul_one, zero_add] |
| exact (Real.continuousAt_log hb).continuousWithinAt |
| · have hx1 : x = 1 := h |
| simp only [hx1, mul_one, sub_self, mul_zero, add_zero] |
| exact (Real.continuousAt_log ha).continuousWithinAt |
| rw [CFC.log, ← cfc_projectionReweight P hP a b, |
| ← cfc_comp' Real.log (fun x : ℝ => a * x + b * (1 - x)) P hc (by fun_prop) hP, |
| ← cfc_projectionReweight P hP (Real.log a) (Real.log b)] |
| apply cfc_congr |
| intro x hx |
| rcases hp.spectrum_subset ℝ hx with h | h |
| · have hx0 : x = 0 := h |
| simp [hx0] |
| · have hx1 : x = 1 := h |
| simp [hx1] |
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| end ProjectionLog |
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| end GroupRingDeterminant |
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| end |
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| end OAI |
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