File: Mathlib/Analysis/CstarAlgebra/Spectrum.lean

full_name: «_aux_Mathlib_Analysis_CstarAlgebra_Spectrum___macroRules_term_⋆_1»
code: local postfix:max "⋆" => star
goal: Lean.Macro

full_name: _aux_Mathlib_Analysis_CstarAlgebra_Spectrum___unexpand_Star_star_1
code: local postfix:max "⋆" => star
goal: Lean.PrettyPrinter.Unexpander

full_name: unitary.spectrum_subset_circle
code: theorem unitary.spectrum_subset_circle (u : unitary E) :
    spectrum 𝕜 (u : E) ⊆ Metric.sphere 0 1 
goal: spectrum 𝕜 ↑u ⊆ Metric.sphere 0 1

full_name: spectrum.subset_circle_of_unitary
code: theorem spectrum.subset_circle_of_unitary {u : E} (h : u ∈ unitary E) :
    spectrum 𝕜 u ⊆ Metric.sphere 0 1 
goal: spectrum 𝕜 u ⊆ Metric.sphere 0 1

full_name: _aux_Mathlib_Analysis_CstarAlgebra_Spectrum___unexpand_algebraMap_1
code: local notation "↑ₐ" => algebraMap ℂ A
goal: Lean.PrettyPrinter.Unexpander

full_name: «_aux_Mathlib_Analysis_CstarAlgebra_Spectrum___macroRules_term↑ₐ_1»
code: local notation "↑ₐ" => algebraMap ℂ A
goal: Lean.Macro

full_name: IsSelfAdjoint.spectralRadius_eq_nnnorm
code: theorem IsSelfAdjoint.spectralRadius_eq_nnnorm {a : A} (ha : IsSelfAdjoint a) :
    spectralRadius ℂ a = ‖a‖₊ 
goal: spectralRadius ℂ a = ↑‖a‖₊

full_name: IsSelfAdjoint.toReal_spectralRadius_complex_eq_norm
code: /-- In a C⋆-algebra, the spectral radius of a self-adjoint element is equal to its norm.
See `IsSelfAdjoint.toReal_spectralRadius_eq_norm` for a version involving
`spectralRadius ℝ a`. -/
lemma IsSelfAdjoint.toReal_spectralRadius_complex_eq_norm {a : A} (ha : IsSelfAdjoint a) :
    (spectralRadius ℂ a).toReal = ‖a‖ 
goal: (spectralRadius ℂ a).toReal = ‖a‖

full_name: IsStarNormal.spectralRadius_eq_nnnorm
code: theorem IsStarNormal.spectralRadius_eq_nnnorm (a : A) [IsStarNormal a] :
    spectralRadius ℂ a = ‖a‖₊ 
goal: spectralRadius ℂ a = ↑‖a‖₊

full_name: IsSelfAdjoint.mem_spectrum_eq_re
code: /-- Any element of the spectrum of a selfadjoint is real. -/
theorem IsSelfAdjoint.mem_spectrum_eq_re [StarModule ℂ A] {a : A} (ha : IsSelfAdjoint a) {z : ℂ}
    (hz : z ∈ spectrum ℂ a) : z = z.re 
goal: z = ↑z.re

full_name: selfAdjoint.mem_spectrum_eq_re
code: /-- Any element of the spectrum of a selfadjoint is real. -/
theorem selfAdjoint.mem_spectrum_eq_re [StarModule ℂ A] (a : selfAdjoint A) {z : ℂ}
    (hz : z ∈ spectrum ℂ (a : A)) : z = z.re 
goal: z = ↑z.re

full_name: IsSelfAdjoint.val_re_map_spectrum
code: /-- The spectrum of a selfadjoint is real -/
theorem IsSelfAdjoint.val_re_map_spectrum [StarModule ℂ A] {a : A} (ha : IsSelfAdjoint a) :
    spectrum ℂ a = ((↑) ∘ re '' spectrum ℂ a : Set ℂ) 
goal: spectrum ℂ a = Complex.ofReal' ∘ Complex.re '' spectrum ℂ a

full_name: selfAdjoint.val_re_map_spectrum
code: /-- The spectrum of a selfadjoint is real -/
theorem selfAdjoint.val_re_map_spectrum [StarModule ℂ A] (a : selfAdjoint A) :
    spectrum ℂ (a : A) = ((↑) ∘ re '' spectrum ℂ (a : A) : Set ℂ) 
goal: spectrum ℂ ↑a = Complex.ofReal' ∘ Complex.re '' spectrum ℂ ↑a

full_name: StarAlgHom.nnnorm_apply_le
code: /-- A star algebra homomorphism of complex C⋆-algebras is norm contractive. -/
theorem nnnorm_apply_le (a : A) : ‖(φ a : B)‖₊ ≤ ‖a‖₊ 
goal: ‖φ a‖₊ ≤ ‖a‖₊

full_name: StarAlgHom.norm_apply_le
code: /-- A star algebra homomorphism of complex C⋆-algebras is norm contractive. -/
theorem norm_apply_le (a : A) : ‖(φ a : B)‖ ≤ ‖a‖ 
goal: ‖φ a‖ ≤ ‖a‖

full_name: StarAlgHom.instContinuousLinearMapClassComplex
code: /-- Star algebra homomorphisms between C⋆-algebras are continuous linear maps.
See note [lower instance priority] -/
noncomputable instance (priority := 100) : ContinuousLinearMapClass F ℂ A B 
goal: ContinuousLinearMapClass F ℂ A B

full_name: StarAlgEquiv.nnnorm_map
code: lemma nnnorm_map (φ : F) (a : A) : ‖φ a‖₊ = ‖a‖₊ 
goal: ‖φ a‖₊ = ‖a‖₊

full_name: StarAlgEquiv.norm_map
code: lemma norm_map (φ : F) (a : A) : ‖φ a‖ = ‖a‖ 
goal: ‖φ a‖ = ‖a‖

full_name: StarAlgEquiv.isometry
code: lemma isometry (φ : F) : Isometry φ 
goal: Isometry ⇑φ

full_name: WeakDual.Complex.instStarHomClass
code: /-- This instance is provided instead of `StarAlgHomClass` to avoid type class inference loops.
See note [lower instance priority] -/
noncomputable instance (priority := 100) Complex.instStarHomClass : StarHomClass F A ℂ 
goal: StarHomClass F A ℂ

full_name: AlgHomClass.instStarAlgHomClass
code: /-- This is not an instance to avoid type class inference loops. See
`WeakDual.Complex.instStarHomClass`. -/
lemma _root_.AlgHomClass.instStarAlgHomClass : StarAlgHomClass F ℂ A ℂ 
goal: StarAlgHomClass F ℂ A ℂ

full_name: WeakDual.CharacterSpace.instStarAlgHomClass
code: noncomputable instance instStarAlgHomClass : StarAlgHomClass (characterSpace ℂ A) ℂ A ℂ 
goal: StarAlgHomClass ↑(WeakDual.characterSpace ℂ A) ℂ A ℂ