English
There is a natural map from the character space of the unital subalgebra generated by a to the spectrum of a, given by evaluation on a.
Русский
Существует естественный отображение между пространством характеров единичной подплоскости, порождаемой a, и спектром a, заданное оценкой на a.
LaTeX
$$$\\text{characterSpaceToSpectrum}(a) : \\mathrm{characterSpace}(\\mathbb{C}, \\mathrm{elemental}(\\mathbb{C}, a)) \\to \\mathrm{ spectrum}(\\mathbb{C}, a)$$$
Lean4
/-- The natural map from `characterSpace ℂ (elemental ℂ x)` to `spectrum ℂ x` given
by evaluating `φ` at `x`. This is essentially just evaluation of the `gelfandTransform` of `x`,
but because we want something in `spectrum ℂ x`, as opposed to
`spectrum ℂ ⟨x, elemental.self_mem ℂ x⟩` there is slightly more work to do. -/
@[simps]
noncomputable def characterSpaceToSpectrum (x : A) (φ : characterSpace ℂ (elemental ℂ x)) : spectrum ℂ x
where
val := φ ⟨x, self_mem ℂ x⟩
property := by
simpa only [StarSubalgebra.spectrum_eq (hS := isClosed ℂ x) (a := ⟨x, self_mem ℂ x⟩)] using
AlgHom.apply_mem_spectrum φ ⟨x, self_mem ℂ x⟩
