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ddc84c5 | 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 | import Mathlib.Analysis.Complex.Basic
import Mathlib.Algebra.Module.LinearMap.Defs
import Mathlib.Tactic.NoncommRing
namespace OAI
noncomputable section
open scoped BigOperators
namespace Nonhyperlinear.Diffuseness
/-- Cyclicity equates every polynomial defect moment; no spectral assumption is used. -/
theorem trace_defect_pow {A:Type*} [Ring A] [Algebra ℂ A]
(f:A→ₗ[ℂ]ℂ) (hf:∀a b,f (a*b)=f (b*a)) (a b:A) (n:ℕ) :
f ((1-a*b)^n)=f ((1-b*a)^n) := by
have hsem : SemiconjBy b (1-a*b) (1-b*a) := by
change b*(1-a*b)=(1-b*a)*b
noncomm_ring
induction n with
| zero => simp
| succ n ih =>
have hp:= (hsem.pow_right n).eq
have he : (1-a*b)^(n+1)=(1-a*b)^n-a*(b*(1-a*b)^n) := by
rw [pow_succ']
noncomm_ring
have he' : (1-b*a)^(n+1)=(1-b*a)^n-((1-b*a)^n*b)*a := by
rw [pow_succ]
noncomm_ring
rw [he,he',map_sub,map_sub,ih,hp,hf a]
end Nonhyperlinear.Diffuseness
end
end OAI
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