Normed star rings and algebras

A normed star group is a normed group with a compatible star which is isometric.

A C⋆-ring is a normed star group that is also a ring and that verifies the stronger condition ‖x‖^2 ≤ ‖x⋆ * x‖ for all x (which actually implies equality). If a C⋆-ring is also a star algebra, then it is a C⋆-algebra.

To get a C⋆-algebra E over field 𝕜, use [NormedField 𝕜] [StarRing 𝕜] [NormedRing E] [StarRing E] [CstarRing E] [NormedAlgebra 𝕜 E] [StarModule 𝕜 E].

TODO

• Show that ‖x⋆ * x‖ = ‖x‖^2 is equivalent to ‖x⋆ * x‖ = ‖x⋆‖ * ‖x‖, which is used as the definition of C*-algebras in some sources (e.g. Wikipedia).

class NormedStarGroup (E : Type u_1) [SeminormedAddCommGroup E] [StarAddMonoid E] : Prop

A normed star group is a normed group with a compatible star which is isometric.

• norm_star : ∀ (x : E), ‖star x‖ = ‖x‖

@[simp]

theorem NormedStarGroup.norm_star {E : Type u_1} [SeminormedAddCommGroup E] [StarAddMonoid E] [self : NormedStarGroup E] (x : E) : ‖star x‖ = ‖x‖

@[simp]

theorem nnnorm_star {E : Type u_2} [SeminormedAddCommGroup E] [StarAddMonoid E] [NormedStarGroup E] (x : E) : ‖star x‖₊ = ‖x‖₊

def starNormedAddGroupHom {E : Type u_2} [SeminormedAddCommGroup E] [StarAddMonoid E] [NormedStarGroup E] : NormedAddGroupHom E E

The star map in a normed star group is a normed group homomorphism.

Equations

• starNormedAddGroupHom = let __src := starAddEquiv; { toFun := __src.toFun, map_add' := ⋯, bound' := ⋯ }

theorem star_isometry {E : Type u_2} [SeminormedAddCommGroup E] [StarAddMonoid E] [NormedStarGroup E] : Isometry star

The star map in a normed star group is an isometry

@[instance 100]

instance NormedStarGroup.to_continuousStar {E : Type u_2} [SeminormedAddCommGroup E] [StarAddMonoid E] [NormedStarGroup E] : ContinuousStar E

Equations

• ⋯ = ⋯

instance RingHomIsometric.starRingEnd {E : Type u_2} [NormedCommRing E] [StarRing E] [NormedStarGroup E] : RingHomIsometric (starRingEnd E)

Equations

• ⋯ = ⋯

class CstarRing (E : Type u_4) [NonUnitalNormedRing E] [StarRing E] : Prop

A C*-ring is a normed star ring that satisfies the stronger condition ‖x‖ ^ 2 ≤ ‖x⋆ * x‖ for every x. Note that this condition actually implies equality, as is shown in norm_star_mul_self below.

• norm_mul_self_le : ∀ (x : E), ‖x‖ * ‖x‖ ≤ ‖star x * x‖

theorem CstarRing.norm_mul_self_le {E : Type u_4} [NonUnitalNormedRing E] [StarRing E] [self : CstarRing E] (x : E) : ‖x‖ * ‖x‖ ≤ ‖star x * x‖

instance instCstarRingReal : CstarRing ℝ

Equations

• instCstarRingReal = ⋯

@[instance 100]

instance CstarRing.to_normedStarGroup {E : Type u_2} [NonUnitalNormedRing E] [StarRing E] [CstarRing E] : NormedStarGroup E

In a C*-ring, star preserves the norm.

Equations

• ⋯ = ⋯

theorem CstarRing.norm_star_mul_self {E : Type u_2} [NonUnitalNormedRing E] [StarRing E] [CstarRing E] {x : E} : ‖star x * x‖ = ‖x‖ * ‖x‖

theorem CstarRing.norm_self_mul_star {E : Type u_2} [NonUnitalNormedRing E] [StarRing E] [CstarRing E] {x : E} : ‖x * star x‖ = ‖x‖ * ‖x‖

theorem CstarRing.norm_star_mul_self' {E : Type u_2} [NonUnitalNormedRing E] [StarRing E] [CstarRing E] {x : E} : ‖star x * x‖ = ‖star x‖ * ‖x‖

theorem CstarRing.nnnorm_self_mul_star {E : Type u_2} [NonUnitalNormedRing E] [StarRing E] [CstarRing E] {x : E} : ‖x * star x‖₊ = ‖x‖₊ * ‖x‖₊

theorem CstarRing.nnnorm_star_mul_self {E : Type u_2} [NonUnitalNormedRing E] [StarRing E] [CstarRing E] {x : E} : ‖star x * x‖₊ = ‖x‖₊ * ‖x‖₊

@[simp]

theorem CstarRing.star_mul_self_eq_zero_iff {E : Type u_2} [NonUnitalNormedRing E] [StarRing E] [CstarRing E] (x : E) : star x * x = 0 ↔ x = 0

theorem CstarRing.star_mul_self_ne_zero_iff {E : Type u_2} [NonUnitalNormedRing E] [StarRing E] [CstarRing E] (x : E) : star x * x ≠ 0 ↔ x ≠ 0

@[simp]

theorem CstarRing.mul_star_self_eq_zero_iff {E : Type u_2} [NonUnitalNormedRing E] [StarRing E] [CstarRing E] (x : E) : x * star x = 0 ↔ x = 0

theorem CstarRing.mul_star_self_ne_zero_iff {E : Type u_2} [NonUnitalNormedRing E] [StarRing E] [CstarRing E] (x : E) : x * star x ≠ 0 ↔ x ≠ 0

instance Pi.starRing' {ι : Type u_4} {R : ι → Type u_7} [(i : ι) → NonUnitalNormedRing (R i)] [(i : ι) → StarRing (R i)] : StarRing ((i : ι) → R i)

This instance exists to short circuit type class resolution because of problems with inference involving Π-types.

Equations

• Pi.starRing' = inferInstance

instance Prod.cstarRing {R₁ : Type u_5} {R₂ : Type u_6} [NonUnitalNormedRing R₁] [StarRing R₁] [CstarRing R₁] [NonUnitalNormedRing R₂] [StarRing R₂] [CstarRing R₂] : CstarRing (R₁ × R₂)

Equations

• ⋯ = ⋯

instance Pi.cstarRing {ι : Type u_4} {R : ι → Type u_7} [(i : ι) → NonUnitalNormedRing (R i)] [(i : ι) → StarRing (R i)] [Fintype ι] [∀ (i : ι), CstarRing (R i)] : CstarRing ((i : ι) → R i)

Equations

• ⋯ = ⋯

instance Pi.cstarRing' {ι : Type u_4} {R₁ : Type u_5} [NonUnitalNormedRing R₁] [StarRing R₁] [CstarRing R₁] [Fintype ι] : CstarRing (ι → R₁)

Equations

• ⋯ = ⋯

@[simp]

theorem CstarRing.norm_one {E : Type u_2} [NormedRing E] [StarRing E] [CstarRing E] [Nontrivial E] : ‖1‖ = 1

@[instance 100]

instance CstarRing.instNormOneClassOfNontrivial {E : Type u_2} [NormedRing E] [StarRing E] [CstarRing E] [Nontrivial E] : NormOneClass E

Equations

• ⋯ = ⋯

theorem CstarRing.norm_coe_unitary {E : Type u_2} [NormedRing E] [StarRing E] [CstarRing E] [Nontrivial E] (U : ↥(unitary E)) : ‖↑U‖ = 1

@[simp]

theorem CstarRing.norm_of_mem_unitary {E : Type u_2} [NormedRing E] [StarRing E] [CstarRing E] [Nontrivial E] {U : E} (hU : U ∈ unitary E) : ‖U‖ = 1

@[simp]

theorem CstarRing.norm_coe_unitary_mul {E : Type u_2} [NormedRing E] [StarRing E] [CstarRing E] (U : ↥(unitary E)) (A : E) : ‖↑U * A‖ = ‖A‖

@[simp]

theorem CstarRing.norm_unitary_smul {E : Type u_2} [NormedRing E] [StarRing E] [CstarRing E] (U : ↥(unitary E)) (A : E) : ‖U • A‖ = ‖A‖

theorem CstarRing.norm_mem_unitary_mul {E : Type u_2} [NormedRing E] [StarRing E] [CstarRing E] {U : E} (A : E) (hU : U ∈ unitary E) : ‖U * A‖ = ‖A‖

@[simp]

theorem CstarRing.norm_mul_coe_unitary {E : Type u_2} [NormedRing E] [StarRing E] [CstarRing E] (A : E) (U : ↥(unitary E)) : ‖A * ↑U‖ = ‖A‖

theorem CstarRing.norm_mul_mem_unitary {E : Type u_2} [NormedRing E] [StarRing E] [CstarRing E] (A : E) {U : E} (hU : U ∈ unitary E) : ‖A * U‖ = ‖A‖

theorem IsSelfAdjoint.nnnorm_pow_two_pow {E : Type u_2} [NormedRing E] [StarRing E] [CstarRing E] {x : E} (hx : IsSelfAdjoint x) (n : ℕ) : ‖x ^ 2 ^ n‖₊ = ‖x‖₊ ^ 2 ^ n

theorem selfAdjoint.nnnorm_pow_two_pow {E : Type u_2} [NormedRing E] [StarRing E] [CstarRing E] (x : ↥(selfAdjoint E)) (n : ℕ) : ‖x ^ 2 ^ n‖₊ = ‖x‖₊ ^ 2 ^ n

def starₗᵢ (𝕜 : Type u_1) {E : Type u_2} [CommSemiring 𝕜] [StarRing 𝕜] [SeminormedAddCommGroup E] [StarAddMonoid E] [NormedStarGroup E] [Module 𝕜 E] [StarModule 𝕜 E] : E ≃ₗᵢ⋆[𝕜] E

star bundled as a linear isometric equivalence

Equations

• starₗᵢ 𝕜 = let __src := starAddEquiv; { toFun := __src.toFun, map_add' := ⋯, map_smul' := ⋯, invFun := __src.invFun, left_inv := ⋯, right_inv := ⋯, norm_map' := ⋯ }

@[simp]

theorem coe_starₗᵢ {𝕜 : Type u_1} {E : Type u_2} [CommSemiring 𝕜] [StarRing 𝕜] [SeminormedAddCommGroup E] [StarAddMonoid E] [NormedStarGroup E] [Module 𝕜 E] [StarModule 𝕜 E] : ⇑(starₗᵢ 𝕜) = star

theorem starₗᵢ_apply {𝕜 : Type u_1} {E : Type u_2} [CommSemiring 𝕜] [StarRing 𝕜] [SeminormedAddCommGroup E] [StarAddMonoid E] [NormedStarGroup E] [Module 𝕜 E] [StarModule 𝕜 E] {x : E} : (starₗᵢ 𝕜) x = star x

@[simp]

theorem starₗᵢ_toContinuousLinearEquiv {𝕜 : Type u_1} {E : Type u_2} [CommSemiring 𝕜] [StarRing 𝕜] [SeminormedAddCommGroup E] [StarAddMonoid E] [NormedStarGroup E] [Module 𝕜 E] [StarModule 𝕜 E] : (starₗᵢ 𝕜).toContinuousLinearEquiv = starL 𝕜

instance StarSubalgebra.toNormedAlgebra {𝕜 : Type u_4} {A : Type u_5} [NormedField 𝕜] [StarRing 𝕜] [SeminormedRing A] [StarRing A] [NormedAlgebra 𝕜 A] [StarModule 𝕜 A] (S : StarSubalgebra 𝕜 A) : NormedAlgebra 𝕜 ↥S

Equations

• S.toNormedAlgebra = NormedAlgebra.induced 𝕜 (↥S) A S.subtype

instance StarSubalgebra.to_cstarRing {R : Type u_4} {A : Type u_5} [CommRing R] [StarRing R] [NormedRing A] [StarRing A] [CstarRing A] [Algebra R A] [StarModule R A] (S : StarSubalgebra R A) : CstarRing ↥S

Equations

• ⋯ = ⋯