English
Noncomputable construction of the Gelfand–Mazur isomorphism between ℂ and a complex Banach division algebra A, under the given hypothesis hA.
Русский
Непcomputable построение изоморфизма Гельфанда-Мазура между ℂ и комплексной банаховой деловой алгеброй A при заданном предположении hA.
LaTeX
$$$\\text{GM\_Isomorphism}(A,hA) : \\mathbb{C} \\cong_{\\mathbb{C}} A$$$
Lean4
/-- **Gelfand-Mazur theorem**: For a complex Banach division algebra, the natural `algebraMap ℂ A`
is an algebra isomorphism whose inverse is given by selecting the (unique) element of
`spectrum ℂ a`. In addition, `algebraMap_isometry` guarantees this map is an isometry.
Note: because `NormedDivisionRing` requires the field `norm_mul : ∀ a b, ‖a * b‖ = ‖a‖ * ‖b‖`, we
don't use this type class and instead opt for a `NormedRing` in which the nonzero elements are
precisely the units. This allows for the application of this isomorphism in broader contexts, e.g.,
to the quotient of a complex Banach algebra by a maximal ideal. In the case when `A` is actually a
`NormedDivisionRing`, one may fill in the argument `hA` with the lemma `isUnit_iff_ne_zero`. -/
@[simps]
noncomputable def _root_.NormedRing.algEquivComplexOfComplete (hA : ∀ {a : A}, IsUnit a ↔ a ≠ 0) [CompleteSpace A] :
ℂ ≃ₐ[ℂ] A :=
let nt : Nontrivial A := ⟨⟨1, 0, hA.mp ⟨⟨1, 1, mul_one _, mul_one _⟩, rfl⟩⟩⟩
{ Algebra.ofId ℂ A with
toFun := algebraMap ℂ A
invFun := fun a => (@spectrum.nonempty _ _ _ _ nt a).some
left_inv := fun z => by
simpa only [@scalar_eq _ _ _ _ _ nt _] using (@spectrum.nonempty _ _ _ _ nt <| algebraMap ℂ A z).some_mem
right_inv := fun a => algebraMap_eq_of_mem (@hA) (@spectrum.nonempty _ _ _ _ nt a).some_mem }
