Theorem: spectrum_star_mul_self_of_isStarNormal
File: Mathlib/Analysis/NormedSpace/Star/ContinuousFunctionalCalculus.lean

tactic: rcases subsingleton_or_nontrivial A with ⟨⟩
state_before: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
⊢ spectrum ℂ (star a * a) ⊆ Set.Icc 0 ↑‖star a * a‖
state_after: case inl
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
h✝ : Subsingleton A
⊢ spectrum ℂ (star a * a) ⊆ Set.Icc 0 ↑‖star a * a‖

case inr
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
h✝ : Nontrivial A
⊢ spectrum ℂ (star a * a) ⊆ Set.Icc 0 ↑‖star a * a‖

tactic: simp only [spectrum.of_subsingleton, Set.empty_subset]
state_before: case inl
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
h✝ : Subsingleton A
⊢ spectrum ℂ (star a * a) ⊆ Set.Icc 0 ↑‖star a * a‖
state_after: no goals

tactic: set a' : elementalStarAlgebra ℂ a := ⟨a, self_mem ℂ a⟩
state_before: case inr
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
h✝ : Nontrivial A
⊢ spectrum ℂ (star a * a) ⊆ Set.Icc 0 ↑‖star a * a‖
state_after: case inr
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
h✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
⊢ spectrum ℂ (star a * a) ⊆ Set.Icc 0 ↑‖star a * a‖

tactic: refine (spectrum.subset_starSubalgebra (star a' * a')).trans ?_
state_before: case inr
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
h✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
⊢ spectrum ℂ (star a * a) ⊆ Set.Icc 0 ↑‖star a * a‖
state_after: case inr
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
h✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
⊢ spectrum ℂ (star a' * a') ⊆ Set.Icc 0 ↑‖star a * a‖

tactic: rw [← spectrum.gelfandTransform_eq (star a' * a'), ContinuousMap.spectrum_eq_range]
state_before: case inr
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
h✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
⊢ spectrum ℂ (star a' * a') ⊆ Set.Icc 0 ↑‖star a * a‖
state_after: case inr
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
h✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
⊢ Set.range ⇑((gelfandTransform ℂ ↥(elementalStarAlgebra ℂ a)) (star a' * a')) ⊆ Set.Icc 0 ↑‖star a * a‖

tactic: rintro - ⟨φ, rfl⟩
state_before: case inr
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
h✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
⊢ Set.range ⇑((gelfandTransform ℂ ↥(elementalStarAlgebra ℂ a)) (star a' * a')) ⊆ Set.Icc 0 ↑‖star a * a‖
state_after: case inr.intro
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
h✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
φ : ↑(characterSpace ℂ ↥(elementalStarAlgebra ℂ a))
⊢ ((gelfandTransform ℂ ↥(elementalStarAlgebra ℂ a)) (star a' * a')) φ ∈ Set.Icc 0 ↑‖star a * a‖

tactic: rw [gelfandTransform_apply_apply ℂ _ (star a' * a') φ, map_mul φ, map_star φ]
state_before: case inr.intro
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
h✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
φ : ↑(characterSpace ℂ ↥(elementalStarAlgebra ℂ a))
⊢ ((gelfandTransform ℂ ↥(elementalStarAlgebra ℂ a)) (star a' * a')) φ ∈ Set.Icc 0 ↑‖star a * a‖
state_after: case inr.intro
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
h✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
φ : ↑(characterSpace ℂ ↥(elementalStarAlgebra ℂ a))
⊢ star (φ a') * φ a' ∈ Set.Icc 0 ↑‖star a * a‖

tactic: rw [Complex.eq_coe_norm_of_nonneg (star_mul_self_nonneg _), ← map_star, ← map_mul]
state_before: case inr.intro
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
h✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
φ : ↑(characterSpace ℂ ↥(elementalStarAlgebra ℂ a))
⊢ star (φ a') * φ a' ∈ Set.Icc 0 ↑‖star a * a‖
state_after: case inr.intro
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
h✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
φ : ↑(characterSpace ℂ ↥(elementalStarAlgebra ℂ a))
⊢ ↑‖φ (star a' * a')‖ ∈ Set.Icc 0 ↑‖star a * a‖

tactic: exact ⟨by positivity, Complex.real_le_real.2 (AlgHom.norm_apply_le_self φ (star a' * a'))⟩
state_before: case inr.intro
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
h✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
φ : ↑(characterSpace ℂ ↥(elementalStarAlgebra ℂ a))
⊢ ↑‖φ (star a' * a')‖ ∈ Set.Icc 0 ↑‖star a * a‖
state_after: no goals

tactic: positivity
state_before: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
h✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
φ : ↑(characterSpace ℂ ↥(elementalStarAlgebra ℂ a))
⊢ 0 ≤ ↑‖φ (star a' * a')‖
state_after: no goals