File size: 1,909 Bytes
67aee1d | 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 | import OAI.Analysis.GroupDeterminants.RegularAdjoint
namespace OAI
universe uA
noncomputable section
open scoped BigOperators
namespace GroupRingDeterminant
section ProjectionLog
variable {A : Type uA} [CStarAlgebra A]
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]
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]
end ProjectionLog
end GroupRingDeterminant
end
end OAI
|