Morphisms of star algebras

This file defines morphisms between R-algebras (unital or non-unital) A and B where both A and B are equipped with a star operation. These morphisms, namely StarAlgHom and NonUnitalStarAlgHom are direct extensions of their non-starred counterparts with a field map_star which guarantees they preserve the star operation. We keep the type classes as generic as possible, in keeping with the definition of NonUnitalAlgHom in the non-unital case. In this file, we only assume Star unless we want to talk about the zero map as a NonUnitalStarAlgHom, in which case we need StarAddMonoid. Note that the scalar ring R is not required to have a star operation, nor do we need StarRing or StarModule structures on A and B.

As with NonUnitalAlgHom, in the non-unital case the multiplications are not assumed to be associative or unital, or even to be compatible with the scalar actions. In a typical application, the operations will satisfy compatibility conditions making them into algebras (albeit possibly non-associative and/or non-unital) but such conditions are not required here for the definitions.

The primary impetus for defining these types is that they constitute the morphisms in the categories of unital C⋆-algebras (with StarAlgHoms) and of C⋆-algebras (with NonUnitalStarAlgHoms).

Main definitions

• NonUnitalStarAlgHom
• StarAlgHom

Tags

non-unital, algebra, morphism, star

Non-unital star algebra homomorphisms

structure NonUnitalStarAlgHom (R : Type u_1) (A : Type u_2) (B : Type u_3) [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] extends A →ₙₐ[R] B : Type (max u_2 u_3)

A non-unital ⋆-algebra homomorphism is a non-unital algebra homomorphism between non-unital R-algebras A and B equipped with a star operation, and this homomorphism is also star-preserving.

• toFun : A → B
• map_smul' (m : R) (x : A) : self.toFun (m • x) = (MonoidHom.id R) m • self.toFun x
• map_zero' : self.toFun 0 = 0
• map_add' (x y : A) : self.toFun (x + y) = self.toFun x + self.toFun y
• map_mul' (x y : A) : self.toFun (x * y) = self.toFun x * self.toFun y
• map_star' (a : A) : self.toFun (star a) = star (self.toFun a)

  By definition, a non-unital ⋆-algebra homomorphism preserves the star operation.

def «term_→⋆ₙₐ_» : Lean.TrailingParserDescr

A non-unital ⋆-algebra homomorphism is a non-unital algebra homomorphism between non-unital R-algebras A and B equipped with a star operation, and this homomorphism is also star-preserving.

Equations

• «term_→⋆ₙₐ_» = Lean.ParserDescr.trailingNode `«term_→⋆ₙₐ_» 25 26 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " →⋆ₙₐ ") (Lean.ParserDescr.cat `term 25))

def «term_→⋆ₙₐ[_]_» : Lean.TrailingParserDescr

A non-unital ⋆-algebra homomorphism is a non-unital algebra homomorphism between non-unital R-algebras A and B equipped with a star operation, and this homomorphism is also star-preserving.

Equations

• One or more equations did not get rendered due to their size.

def NonUnitalStarAlgHomClass.toNonUnitalStarAlgHom {F : Type u_1} {R : Type u_2} {A : Type u_3} {B : Type u_4} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [StarHomClass F A B] (f : F) : A →⋆ₙₐ[R] B

Turn an element of a type F satisfying NonUnitalAlgHomClass F R A B and StarHomClass F A B into an actual NonUnitalStarAlgHom. This is declared as the default coercion from F to A →⋆ₙₐ[R] B.

Equations

• ↑f = { toNonUnitalAlgHom := NonUnitalAlgHomClass.toNonUnitalAlgHom f, map_star' := ⋯ }

instance NonUnitalStarAlgHomClass.instCoeTCNonUnitalStarAlgHomOfStarHomClass {F : Type u_1} {R : Type u_2} {A : Type u_3} {B : Type u_4} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [StarHomClass F A B] : CoeTC F (A →⋆ₙₐ[R] B)

Equations

• NonUnitalStarAlgHomClass.instCoeTCNonUnitalStarAlgHomOfStarHomClass = { coe := NonUnitalStarAlgHomClass.toNonUnitalStarAlgHom }

instance NonUnitalStarAlgHomClass.instNonUnitalStarRingHomClassOfStarHomClass {F : Type u_1} {R : Type u_2} {A : Type u_3} {B : Type u_4} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [StarHomClass F A B] : NonUnitalStarRingHomClass F A B

instance NonUnitalStarAlgHom.instFunLike {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] : FunLike (A →⋆ₙₐ[R] B) A B

Equations

• NonUnitalStarAlgHom.instFunLike = { coe := fun (f : A →⋆ₙₐ[R] B) => f.toFun, coe_injective' := ⋯ }

instance NonUnitalStarAlgHom.instNonUnitalAlgHomClass {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] : NonUnitalAlgHomClass (A →⋆ₙₐ[R] B) R A B

instance NonUnitalStarAlgHom.instStarHomClass {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] : StarHomClass (A →⋆ₙₐ[R] B) A B

def NonUnitalStarAlgHom.Simps.apply {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] (f : A →⋆ₙₐ[R] B) : A → B

See Note [custom simps projection]

Equations

• NonUnitalStarAlgHom.Simps.apply f = ⇑f

@[simp]

theorem NonUnitalStarAlgHom.coe_coe {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] {F : Type u_6} [FunLike F A B] [NonUnitalAlgHomClass F R A B] [StarHomClass F A B] (f : F) : ⇑↑f = ⇑f

@[simp]

theorem NonUnitalStarAlgHom.coe_toNonUnitalAlgHom {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] {f : A →⋆ₙₐ[R] B} : ⇑f.toNonUnitalAlgHom = ⇑f

theorem NonUnitalStarAlgHom.ext {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] {f g : A →⋆ₙₐ[R] B} (h : ∀ (x : A), f x = g x) : f = g

theorem NonUnitalStarAlgHom.ext_iff {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] {f g : A →⋆ₙₐ[R] B} : f = g ↔ ∀ (x : A), f x = g x

def NonUnitalStarAlgHom.copy {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] (f : A →⋆ₙₐ[R] B) (f' : A → B) (h : f' = ⇑f) : A →⋆ₙₐ[R] B

Copy of a NonUnitalStarAlgHom with a new toFun equal to the old one. Useful to fix definitional equalities.

Equations

• f.copy f' h = { toFun := f', map_smul' := ⋯, map_zero' := ⋯, map_add' := ⋯, map_mul' := ⋯, map_star' := ⋯ }

@[simp]

theorem NonUnitalStarAlgHom.coe_copy {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] (f : A →⋆ₙₐ[R] B) (f' : A → B) (h : f' = ⇑f) : ⇑(f.copy f' h) = f'

theorem NonUnitalStarAlgHom.copy_eq {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] (f : A →⋆ₙₐ[R] B) (f' : A → B) (h : f' = ⇑f) : f.copy f' h = f

@[simp]

theorem NonUnitalStarAlgHom.coe_mk {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] (f : A → B) (h₁ : ∀ (m : R) (x : A), f (m • x) = (MonoidHom.id R) m • f x) (h₂ : { toFun := f, map_smul' := h₁ }.toFun 0 = 0) (h₃ : ∀ (x y : A), { toFun := f, map_smul' := h₁ }.toFun (x + y) = { toFun := f, map_smul' := h₁ }.toFun x + { toFun := f, map_smul' := h₁ }.toFun y) (h₄ : ∀ (x y : A), { toFun := f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃ }.toFun (x * y) = { toFun := f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃ }.toFun x * { toFun := f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃ }.toFun y) (h₅ : ∀ (a : A), { toFun := f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃, map_mul' := h₄ }.toFun (star a) = star ({ toFun := f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃, map_mul' := h₄ }.toFun a)) : ⇑{ toFun := f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃, map_mul' := h₄, map_star' := h₅ } = f

@[simp]

theorem NonUnitalStarAlgHom.coe_mk' {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] (f : A →ₙₐ[R] B) (h : ∀ (a : A), f.toFun (star a) = star (f.toFun a)) : ⇑{ toNonUnitalAlgHom := f, map_star' := h } = ⇑f

@[simp]

theorem NonUnitalStarAlgHom.mk_coe {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] (f : A →⋆ₙₐ[R] B) (h₁ : ∀ (m : R) (x : A), f (m • x) = (MonoidHom.id R) m • f x) (h₂ : { toFun := ⇑f, map_smul' := h₁ }.toFun 0 = 0) (h₃ : ∀ (x y : A), { toFun := ⇑f, map_smul' := h₁ }.toFun (x + y) = { toFun := ⇑f, map_smul' := h₁ }.toFun x + { toFun := ⇑f, map_smul' := h₁ }.toFun y) (h₄ : ∀ (x y : A), { toFun := ⇑f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃ }.toFun (x * y) = { toFun := ⇑f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃ }.toFun x * { toFun := ⇑f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃ }.toFun y) (h₅ : ∀ (a : A), { toFun := ⇑f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃, map_mul' := h₄ }.toFun (star a) = star ({ toFun := ⇑f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃, map_mul' := h₄ }.toFun a)) : { toFun := ⇑f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃, map_mul' := h₄, map_star' := h₅ } = f

def NonUnitalStarAlgHom.id (R : Type u_1) (A : Type u_2) [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] : A →⋆ₙₐ[R] A

The identity as a non-unital ⋆-algebra homomorphism.

Equations

• NonUnitalStarAlgHom.id R A = { toNonUnitalAlgHom := 1, map_star' := ⋯ }

@[simp]

theorem NonUnitalStarAlgHom.coe_id (R : Type u_1) (A : Type u_2) [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] : ⇑(NonUnitalStarAlgHom.id R A) = id

def NonUnitalStarAlgHom.comp {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] [NonUnitalNonAssocSemiring C] [DistribMulAction R C] [Star C] (f : B →⋆ₙₐ[R] C) (g : A →⋆ₙₐ[R] B) : A →⋆ₙₐ[R] C

The composition of non-unital ⋆-algebra homomorphisms, as a non-unital ⋆-algebra homomorphism.

Equations

• f.comp g = { toNonUnitalAlgHom := f.comp g.toNonUnitalAlgHom, map_star' := ⋯ }

@[simp]

theorem NonUnitalStarAlgHom.coe_comp {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] [NonUnitalNonAssocSemiring C] [DistribMulAction R C] [Star C] (f : B →⋆ₙₐ[R] C) (g : A →⋆ₙₐ[R] B) : ⇑(f.comp g) = ⇑f ∘ ⇑g

@[simp]

theorem NonUnitalStarAlgHom.comp_apply {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] [NonUnitalNonAssocSemiring C] [DistribMulAction R C] [Star C] (f : B →⋆ₙₐ[R] C) (g : A →⋆ₙₐ[R] B) (a : A) : (f.comp g) a = f (g a)

@[simp]

theorem NonUnitalStarAlgHom.comp_assoc {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} {D : Type u_5} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] [NonUnitalNonAssocSemiring C] [DistribMulAction R C] [Star C] [NonUnitalNonAssocSemiring D] [DistribMulAction R D] [Star D] (f : C →⋆ₙₐ[R] D) (g : B →⋆ₙₐ[R] C) (h : A →⋆ₙₐ[R] B) : (f.comp g).comp h = f.comp (g.comp h)

@[simp]

theorem NonUnitalStarAlgHom.id_comp {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] (f : A →⋆ₙₐ[R] B) : (NonUnitalStarAlgHom.id R B).comp f = f

@[simp]

theorem NonUnitalStarAlgHom.comp_id {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] (f : A →⋆ₙₐ[R] B) : f.comp (NonUnitalStarAlgHom.id R A) = f

instance NonUnitalStarAlgHom.instMonoid {R : Type u_1} {A : Type u_2} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] : Monoid (A →⋆ₙₐ[R] A)

Equations

• NonUnitalStarAlgHom.instMonoid = { mul := NonUnitalStarAlgHom.comp, mul_assoc := ⋯, one := NonUnitalStarAlgHom.id R A, one_mul := ⋯, mul_one := ⋯, npow_zero := ⋯, npow_succ := ⋯ }

@[simp]

theorem NonUnitalStarAlgHom.coe_one {R : Type u_1} {A : Type u_2} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] : ⇑1 = id

theorem NonUnitalStarAlgHom.one_apply {R : Type u_1} {A : Type u_2} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] (a : A) : 1 a = a

instance NonUnitalStarAlgHom.instZero {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [StarAddMonoid A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [StarAddMonoid B] : Zero (A →⋆ₙₐ[R] B)

Equations

• NonUnitalStarAlgHom.instZero = { zero := let __src := 0; { toNonUnitalAlgHom := __src, map_star' := ⋯ } }

instance NonUnitalStarAlgHom.instInhabited {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [StarAddMonoid A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [StarAddMonoid B] : Inhabited (A →⋆ₙₐ[R] B)

Equations

• NonUnitalStarAlgHom.instInhabited = { default := 0 }

instance NonUnitalStarAlgHom.instMonoidWithZero {R : Type u_1} {A : Type u_2} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [StarAddMonoid A] : MonoidWithZero (A →⋆ₙₐ[R] A)

Equations

• NonUnitalStarAlgHom.instMonoidWithZero = { toMonoid := inferInstanceAs (Monoid (A →⋆ₙₐ[R] A)), toZero := inferInstanceAs (Zero (A →⋆ₙₐ[R] A)), zero_mul := ⋯, mul_zero := ⋯ }

@[simp]

theorem NonUnitalStarAlgHom.coe_zero {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [StarAddMonoid A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [StarAddMonoid B] : ⇑0 = 0

theorem NonUnitalStarAlgHom.zero_apply {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [StarAddMonoid A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [StarAddMonoid B] (a : A) : 0 a = 0

def NonUnitalStarAlgHom.restrictScalars (R : Type u_1) {S : Type u_2} {A : Type u_3} {B : Type u_4} [Monoid R] [Monoid S] [Star A] [Star B] [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [MulAction R S] [DistribMulAction S A] [DistribMulAction S B] [DistribMulAction R A] [DistribMulAction R B] [IsScalarTower R S A] [IsScalarTower R S B] (f : A →⋆ₙₐ[S] B) : A →⋆ₙₐ[R] B

If a monoid R acts on another monoid S, then a non-unital star algebra homomorphism over S can be viewed as a non-unital star algebra homomorphism over R.

Equations

• NonUnitalStarAlgHom.restrictScalars R f = { toNonUnitalAlgHom := NonUnitalAlgHom.restrictScalars R (NonUnitalAlgHomClass.toNonUnitalAlgHom f), map_star' := ⋯ }

@[simp]

theorem NonUnitalStarAlgHom.restrictScalars_apply (R : Type u_1) {S : Type u_2} {A : Type u_3} {B : Type u_4} [Monoid R] [Monoid S] [Star A] [Star B] [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [MulAction R S] [DistribMulAction S A] [DistribMulAction S B] [DistribMulAction R A] [DistribMulAction R B] [IsScalarTower R S A] [IsScalarTower R S B] (f : A →⋆ₙₐ[S] B) (x : A) : (restrictScalars R f) x = f x

theorem NonUnitalStarAlgHom.coe_restrictScalars (R : Type u_1) {S : Type u_2} {A : Type u_3} {B : Type u_4} [Monoid R] [Monoid S] [Star A] [Star B] [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [MulAction R S] [DistribMulAction S A] [DistribMulAction S B] [DistribMulAction R A] [DistribMulAction R B] [IsScalarTower R S A] [IsScalarTower R S B] (f : A →⋆ₙₐ[S] B) : ↑(restrictScalars R f) = ↑f

theorem NonUnitalStarAlgHom.coe_restrictScalars' (R : Type u_1) {S : Type u_2} {A : Type u_3} {B : Type u_4} [Monoid R] [Monoid S] [Star A] [Star B] [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [MulAction R S] [DistribMulAction S A] [DistribMulAction S B] [DistribMulAction R A] [DistribMulAction R B] [IsScalarTower R S A] [IsScalarTower R S B] (f : A →⋆ₙₐ[S] B) : ⇑(restrictScalars R f) = ⇑f

theorem NonUnitalStarAlgHom.restrictScalars_injective (R : Type u_1) {S : Type u_2} {A : Type u_3} {B : Type u_4} [Monoid R] [Monoid S] [Star A] [Star B] [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [MulAction R S] [DistribMulAction S A] [DistribMulAction S B] [DistribMulAction R A] [DistribMulAction R B] [IsScalarTower R S A] [IsScalarTower R S B] : Function.Injective (restrictScalars R)

Unital star algebra homomorphisms

structure StarAlgHom (R : Type u_1) (A : Type u_2) (B : Type u_3) [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] extends A →ₐ[R] B : Type (max u_2 u_3)

A ⋆-algebra homomorphism is an algebra homomorphism between R-algebras A and B equipped with a star operation, and this homomorphism is also star-preserving.

• toFun : A → B
• map_one' : (↑↑self.toRingHom).toFun 1 = 1
• map_mul' (x y : A) : (↑↑self.toRingHom).toFun (x * y) = (↑↑self.toRingHom).toFun x * (↑↑self.toRingHom).toFun y
• map_zero' : (↑↑self.toRingHom).toFun 0 = 0
• map_add' (x y : A) : (↑↑self.toRingHom).toFun (x + y) = (↑↑self.toRingHom).toFun x + (↑↑self.toRingHom).toFun y
• commutes' (r : R) : (↑↑self.toRingHom).toFun ((algebraMap R A) r) = (algebraMap R B) r
• map_star' (x : A) : (↑↑self.toRingHom).toFun (star x) = star ((↑↑self.toRingHom).toFun x)

  By definition, a ⋆-algebra homomorphism preserves the star operation.

def «term_→⋆ₐ_» : Lean.TrailingParserDescr

A ⋆-algebra homomorphism is an algebra homomorphism between R-algebras A and B equipped with a star operation, and this homomorphism is also star-preserving.

Equations

• «term_→⋆ₐ_» = Lean.ParserDescr.trailingNode `«term_→⋆ₐ_» 25 26 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " →⋆ₐ ") (Lean.ParserDescr.cat `term 25))

def «term_→⋆ₐ[_]_» : Lean.TrailingParserDescr

A ⋆-algebra homomorphism is an algebra homomorphism between R-algebras A and B equipped with a star operation, and this homomorphism is also star-preserving.

Equations

• One or more equations did not get rendered due to their size.

def StarAlgHomClass.toStarAlgHom {F : Type u_1} {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] [FunLike F A B] [AlgHomClass F R A B] [StarHomClass F A B] (f : F) : A →⋆ₐ[R] B

Turn an element of a type F satisfying AlgHomClass F R A B and StarHomClass F A B into an actual StarAlgHom. This is declared as the default coercion from F to A →⋆ₐ[R] B.

Equations

• ↑f = { toAlgHom := ↑f, map_star' := ⋯ }

instance StarAlgHomClass.instCoeTCStarAlgHom {F : Type u_1} {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] [FunLike F A B] [AlgHomClass F R A B] [StarHomClass F A B] : CoeTC F (A →⋆ₐ[R] B)

Equations

• StarAlgHomClass.instCoeTCStarAlgHom = { coe := StarAlgHomClass.toStarAlgHom }

instance StarAlgHom.instFunLike {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] : FunLike (A →⋆ₐ[R] B) A B

Equations

• StarAlgHom.instFunLike = { coe := fun (f : A →⋆ₐ[R] B) => (↑↑f.toRingHom).toFun, coe_injective' := ⋯ }

instance StarAlgHom.instAlgHomClass {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] : AlgHomClass (A →⋆ₐ[R] B) R A B

instance StarAlgHom.instStarHomClass {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] : StarHomClass (A →⋆ₐ[R] B) A B

@[simp]

theorem StarAlgHom.coe_coe {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] {F : Type u_7} [FunLike F A B] [AlgHomClass F R A B] [StarHomClass F A B] (f : F) : ⇑↑f = ⇑f

def StarAlgHom.Simps.apply {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] (f : A →⋆ₐ[R] B) : A → B

See Note [custom simps projection]

Equations

• StarAlgHom.Simps.apply f = ⇑f

@[simp]

theorem StarAlgHom.coe_toAlgHom {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] {f : A →⋆ₐ[R] B} : ⇑f.toAlgHom = ⇑f

theorem StarAlgHom.ext {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] {f g : A →⋆ₐ[R] B} (h : ∀ (x : A), f x = g x) : f = g

theorem StarAlgHom.ext_iff {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] {f g : A →⋆ₐ[R] B} : f = g ↔ ∀ (x : A), f x = g x

def StarAlgHom.copy {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] (f : A →⋆ₐ[R] B) (f' : A → B) (h : f' = ⇑f) : A →⋆ₐ[R] B

Copy of a StarAlgHom with a new toFun equal to the old one. Useful to fix definitional equalities.

Equations

• f.copy f' h = { toFun := f', map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯, commutes' := ⋯, map_star' := ⋯ }

@[simp]

theorem StarAlgHom.coe_copy {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] (f : A →⋆ₐ[R] B) (f' : A → B) (h : f' = ⇑f) : ⇑(f.copy f' h) = f'

theorem StarAlgHom.copy_eq {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] (f : A →⋆ₐ[R] B) (f' : A → B) (h : f' = ⇑f) : f.copy f' h = f

@[simp]

theorem StarAlgHom.coe_mk {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] (f : A → B) (h₁ : f 1 = 1) (h₂ : ∀ (x y : A), { toFun := f, map_one' := h₁ }.toFun (x * y) = { toFun := f, map_one' := h₁ }.toFun x * { toFun := f, map_one' := h₁ }.toFun y) (h₃ : (↑{ toFun := f, map_one' := h₁, map_mul' := h₂ }).toFun 0 = 0) (h₄ : ∀ (x y : A), (↑{ toFun := f, map_one' := h₁, map_mul' := h₂ }).toFun (x + y) = (↑{ toFun := f, map_one' := h₁, map_mul' := h₂ }).toFun x + (↑{ toFun := f, map_one' := h₁, map_mul' := h₂ }).toFun y) (h₅ : ∀ (r : R), (↑↑{ toFun := f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄ }).toFun ((algebraMap R A) r) = (algebraMap R B) r) (h₆ : ∀ (x : A), (↑↑{ toFun := f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄, commutes' := h₅ }.toRingHom).toFun (star x) = star ((↑↑{ toFun := f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄, commutes' := h₅ }.toRingHom).toFun x)) : ⇑{ toFun := f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄, commutes' := h₅, map_star' := h₆ } = f

@[simp]

theorem StarAlgHom.coe_mk' {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] (f : A →ₐ[R] B) (h : ∀ (x : A), (↑↑f.toRingHom).toFun (star x) = star ((↑↑f.toRingHom).toFun x)) : ⇑{ toAlgHom := f, map_star' := h } = ⇑f

@[simp]

theorem StarAlgHom.mk_coe {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] (f : A →⋆ₐ[R] B) (h₁ : f 1 = 1) (h₂ : ∀ (x y : A), { toFun := ⇑f, map_one' := h₁ }.toFun (x * y) = { toFun := ⇑f, map_one' := h₁ }.toFun x * { toFun := ⇑f, map_one' := h₁ }.toFun y) (h₃ : (↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂ }).toFun 0 = 0) (h₄ : ∀ (x y : A), (↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂ }).toFun (x + y) = (↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂ }).toFun x + (↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂ }).toFun y) (h₅ : ∀ (r : R), (↑↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄ }).toFun ((algebraMap R A) r) = (algebraMap R B) r) (h₆ : ∀ (x : A), (↑↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄, commutes' := h₅ }.toRingHom).toFun (star x) = star ((↑↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄, commutes' := h₅ }.toRingHom).toFun x)) : { toFun := ⇑f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄, commutes' := h₅, map_star' := h₆ } = f

def StarAlgHom.id (R : Type u_2) (A : Type u_3) [CommSemiring R] [Semiring A] [Algebra R A] [Star A] : A →⋆ₐ[R] A

The identity as a StarAlgHom.

Equations

• StarAlgHom.id R A = { toAlgHom := AlgHom.id R A, map_star' := ⋯ }

@[simp]

theorem StarAlgHom.coe_id (R : Type u_2) (A : Type u_3) [CommSemiring R] [Semiring A] [Algebra R A] [Star A] : ⇑(StarAlgHom.id R A) = id

def StarAlgHom.ofId (R : Type u_7) (A : Type u_8) [CommSemiring R] [StarRing R] [Semiring A] [StarMul A] [Algebra R A] [StarModule R A] : R →⋆ₐ[R] A

algebraMap R A as a StarAlgHom when A is a star algebra over R.

Equations

• StarAlgHom.ofId R A = { toFun := ⇑(algebraMap R A), map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯, commutes' := ⋯, map_star' := ⋯ }

@[simp]

theorem StarAlgHom.ofId_apply (R : Type u_7) (A : Type u_8) [CommSemiring R] [StarRing R] [Semiring A] [StarMul A] [Algebra R A] [StarModule R A] (a : R) : (ofId R A) a = (algebraMap R A) a

instance StarAlgHom.instInhabited {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] : Inhabited (A →⋆ₐ[R] A)

Equations

• StarAlgHom.instInhabited = { default := StarAlgHom.id R A }

def StarAlgHom.comp {R : Type u_2} {A : Type u_3} {B : Type u_4} {C : Type u_5} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] [Semiring C] [Algebra R C] [Star C] (f : B →⋆ₐ[R] C) (g : A →⋆ₐ[R] B) : A →⋆ₐ[R] C

The composition of ⋆-algebra homomorphisms, as a ⋆-algebra homomorphism.

Equations

• f.comp g = { toAlgHom := f.comp g.toAlgHom, map_star' := ⋯ }

@[simp]

theorem StarAlgHom.coe_comp {R : Type u_2} {A : Type u_3} {B : Type u_4} {C : Type u_5} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] [Semiring C] [Algebra R C] [Star C] (f : B →⋆ₐ[R] C) (g : A →⋆ₐ[R] B) : ⇑(f.comp g) = ⇑f ∘ ⇑g

@[simp]

theorem StarAlgHom.comp_apply {R : Type u_2} {A : Type u_3} {B : Type u_4} {C : Type u_5} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] [Semiring C] [Algebra R C] [Star C] (f : B →⋆ₐ[R] C) (g : A →⋆ₐ[R] B) (a : A) : (f.comp g) a = f (g a)

@[simp]

theorem StarAlgHom.comp_assoc {R : Type u_2} {A : Type u_3} {B : Type u_4} {C : Type u_5} {D : Type u_6} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] [Semiring C] [Algebra R C] [Star C] [Semiring D] [Algebra R D] [Star D] (f : C →⋆ₐ[R] D) (g : B →⋆ₐ[R] C) (h : A →⋆ₐ[R] B) : (f.comp g).comp h = f.comp (g.comp h)

@[simp]

theorem StarAlgHom.id_comp {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] (f : A →⋆ₐ[R] B) : (StarAlgHom.id R B).comp f = f

@[simp]

theorem StarAlgHom.comp_id {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] (f : A →⋆ₐ[R] B) : f.comp (StarAlgHom.id R A) = f

instance StarAlgHom.instMonoid {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] : Monoid (A →⋆ₐ[R] A)

Equations

• StarAlgHom.instMonoid = { mul := StarAlgHom.comp, mul_assoc := ⋯, one := StarAlgHom.id R A, one_mul := ⋯, mul_one := ⋯, npow_zero := ⋯, npow_succ := ⋯ }

def StarAlgHom.toNonUnitalStarAlgHom {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] (f : A →⋆ₐ[R] B) : A →⋆ₙₐ[R] B

A unital morphism of ⋆-algebras is a NonUnitalStarAlgHom.

Equations

• f.toNonUnitalStarAlgHom = { toFun := (↑↑f.toRingHom).toFun, map_smul' := ⋯, map_zero' := ⋯, map_add' := ⋯, map_mul' := ⋯, map_star' := ⋯ }

@[simp]

theorem StarAlgHom.coe_toNonUnitalStarAlgHom {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] (f : A →⋆ₐ[R] B) : ⇑f.toNonUnitalStarAlgHom = ⇑f

Operations on the product type

Note that this is copied from Algebra.Hom.NonUnitalAlg.

def NonUnitalStarAlgHom.fst (R : Type u_1) (A : Type u_2) (B : Type u_3) [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] : A × B →⋆ₙₐ[R] A

The first projection of a product is a non-unital ⋆-algebra homomorphism.

Equations

• NonUnitalStarAlgHom.fst R A B = { toNonUnitalAlgHom := NonUnitalAlgHom.fst R A B, map_star' := ⋯ }

@[simp]

theorem NonUnitalStarAlgHom.fst_apply (R : Type u_1) (A : Type u_2) (B : Type u_3) [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] (self : A × B) : (fst R A B) self = self.1

def NonUnitalStarAlgHom.snd (R : Type u_1) (A : Type u_2) (B : Type u_3) [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] : A × B →⋆ₙₐ[R] B

The second projection of a product is a non-unital ⋆-algebra homomorphism.

Equations

• NonUnitalStarAlgHom.snd R A B = { toNonUnitalAlgHom := NonUnitalAlgHom.snd R A B, map_star' := ⋯ }

@[simp]

theorem NonUnitalStarAlgHom.snd_apply (R : Type u_1) (A : Type u_2) (B : Type u_3) [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] (self : A × B) : (snd R A B) self = self.2

def NonUnitalStarAlgHom.prod {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] [NonUnitalNonAssocSemiring C] [DistribMulAction R C] [Star C] (f : A →⋆ₙₐ[R] B) (g : A →⋆ₙₐ[R] C) : A →⋆ₙₐ[R] B × C

The Pi.prod of two morphisms is a morphism.

Equations

• f.prod g = { toNonUnitalAlgHom := f.prod g.toNonUnitalAlgHom, map_star' := ⋯ }

@[simp]

theorem NonUnitalStarAlgHom.prod_apply {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] [NonUnitalNonAssocSemiring C] [DistribMulAction R C] [Star C] (f : A →⋆ₙₐ[R] B) (g : A →⋆ₙₐ[R] C) (i : A) : (f.prod g) i = (f i, g i)

theorem NonUnitalStarAlgHom.coe_prod {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] [NonUnitalNonAssocSemiring C] [DistribMulAction R C] [Star C] (f : A →⋆ₙₐ[R] B) (g : A →⋆ₙₐ[R] C) : ⇑(f.prod g) = Pi.prod ⇑f ⇑g

@[simp]

theorem NonUnitalStarAlgHom.fst_prod {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] [NonUnitalNonAssocSemiring C] [DistribMulAction R C] [Star C] (f : A →⋆ₙₐ[R] B) (g : A →⋆ₙₐ[R] C) : (fst R B C).comp (f.prod g) = f

@[simp]

theorem NonUnitalStarAlgHom.snd_prod {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] [NonUnitalNonAssocSemiring C] [DistribMulAction R C] [Star C] (f : A →⋆ₙₐ[R] B) (g : A →⋆ₙₐ[R] C) : (snd R B C).comp (f.prod g) = g

@[simp]

theorem NonUnitalStarAlgHom.prod_fst_snd {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] : (fst R A B).prod (snd R A B) = 1

def NonUnitalStarAlgHom.prodEquiv {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] [NonUnitalNonAssocSemiring C] [DistribMulAction R C] [Star C] : (A →⋆ₙₐ[R] B) × (A →⋆ₙₐ[R] C) ≃ (A →⋆ₙₐ[R] B × C)

Taking the product of two maps with the same domain is equivalent to taking the product of their codomains.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem NonUnitalStarAlgHom.prodEquiv_apply {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] [NonUnitalNonAssocSemiring C] [DistribMulAction R C] [Star C] (f : (A →⋆ₙₐ[R] B) × (A →⋆ₙₐ[R] C)) : prodEquiv f = f.1.prod f.2

@[simp]

theorem NonUnitalStarAlgHom.prodEquiv_symm_apply {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] [NonUnitalNonAssocSemiring C] [DistribMulAction R C] [Star C] (f : A →⋆ₙₐ[R] B × C) : prodEquiv.symm f = ((fst R B C).comp f, (snd R B C).comp f)

def Pi.evalNonUnitalStarAlgHom {ι : Type u_1} (R : Type u_2) (A : ι → Type u_3) (j : ι) [Monoid R] [(i : ι) → NonUnitalNonAssocSemiring (A i)] [(i : ι) → DistribMulAction R (A i)] [(i : ι) → Star (A i)] : ((i : ι) → A i) →⋆ₙₐ[R] A j

Function.eval as a NonUnitalStarAlgHom.

Equations

• Pi.evalNonUnitalStarAlgHom R A j = { toFun := (Pi.evalMulHom A j).toFun, map_smul' := ⋯, map_zero' := ⋯, map_add' := ⋯, map_mul' := ⋯, map_star' := ⋯ }

@[simp]

theorem Pi.evalNonUnitalStarAlgHom_apply {ι : Type u_1} (R : Type u_2) (A : ι → Type u_3) (j : ι) [Monoid R] [(i : ι) → NonUnitalNonAssocSemiring (A i)] [(i : ι) → DistribMulAction R (A i)] [(i : ι) → Star (A i)] (a✝ : (i : ι) → A i) : (evalNonUnitalStarAlgHom R A j) a✝ = (evalMulHom A j).toFun a✝

def Pi.evalStarAlgHom {ι : Type u_1} (R : Type u_2) (A : ι → Type u_3) (j : ι) [CommSemiring R] [(i : ι) → Semiring (A i)] [(i : ι) → Algebra R (A i)] [(i : ι) → Star (A i)] : ((i : ι) → A i) →⋆ₐ[R] A j

Function.eval as a StarAlgHom.

Equations

• Pi.evalStarAlgHom R A j = { toFun := (Pi.evalNonUnitalStarAlgHom R A j).toFun, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯, commutes' := ⋯, map_star' := ⋯ }

@[simp]

theorem Pi.evalStarAlgHom_apply {ι : Type u_1} (R : Type u_2) (A : ι → Type u_3) (j : ι) [CommSemiring R] [(i : ι) → Semiring (A i)] [(i : ι) → Algebra R (A i)] [(i : ι) → Star (A i)] (a✝ : (i : ι) → A i) : (evalStarAlgHom R A j) a✝ = (evalNonUnitalStarAlgHom R A j).toFun a✝

def NonUnitalStarAlgHom.inl (R : Type u_1) (A : Type u_2) (B : Type u_3) [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [StarAddMonoid A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [StarAddMonoid B] : A →⋆ₙₐ[R] A × B

The left injection into a product is a non-unital algebra homomorphism.

Equations

• NonUnitalStarAlgHom.inl R A B = NonUnitalStarAlgHom.prod 1 0

def NonUnitalStarAlgHom.inr (R : Type u_1) (A : Type u_2) (B : Type u_3) [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [StarAddMonoid A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [StarAddMonoid B] : B →⋆ₙₐ[R] A × B

The right injection into a product is a non-unital algebra homomorphism.

Equations

• NonUnitalStarAlgHom.inr R A B = NonUnitalStarAlgHom.prod 0 1

@[simp]

theorem NonUnitalStarAlgHom.coe_inl {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [StarAddMonoid A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [StarAddMonoid B] : ⇑(inl R A B) = fun (x : A) => (x, 0)

theorem NonUnitalStarAlgHom.inl_apply {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [StarAddMonoid A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [StarAddMonoid B] (x : A) : (inl R A B) x = (x, 0)

@[simp]

theorem NonUnitalStarAlgHom.coe_inr {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [StarAddMonoid A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [StarAddMonoid B] : ⇑(inr R A B) = Prod.mk 0

theorem NonUnitalStarAlgHom.inr_apply {R : Type u_1} {A : Type u_2} {B : Type u_3} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [StarAddMonoid A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [StarAddMonoid B] (x : B) : (inr R A B) x = (0, x)

def StarAlgHom.fst (R : Type u_1) (A : Type u_2) (B : Type u_3) [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] : A × B →⋆ₐ[R] A

The first projection of a product is a ⋆-algebra homomorphism.

Equations

• StarAlgHom.fst R A B = { toAlgHom := AlgHom.fst R A B, map_star' := ⋯ }

@[simp]

theorem StarAlgHom.fst_apply (R : Type u_1) (A : Type u_2) (B : Type u_3) [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] (self : A × B) : (fst R A B) self = self.1

def StarAlgHom.snd (R : Type u_1) (A : Type u_2) (B : Type u_3) [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] : A × B →⋆ₐ[R] B

The second projection of a product is a ⋆-algebra homomorphism.

Equations

• StarAlgHom.snd R A B = { toAlgHom := AlgHom.snd R A B, map_star' := ⋯ }

@[simp]

theorem StarAlgHom.snd_apply (R : Type u_1) (A : Type u_2) (B : Type u_3) [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] (self : A × B) : (snd R A B) self = self.2

def StarAlgHom.prod {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] [Semiring C] [Algebra R C] [Star C] (f : A →⋆ₐ[R] B) (g : A →⋆ₐ[R] C) : A →⋆ₐ[R] B × C

The Pi.prod of two morphisms is a morphism.

Equations

• f.prod g = { toAlgHom := f.prod g.toAlgHom, map_star' := ⋯ }

@[simp]

theorem StarAlgHom.prod_apply {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] [Semiring C] [Algebra R C] [Star C] (f : A →⋆ₐ[R] B) (g : A →⋆ₐ[R] C) (x : A) : (f.prod g) x = (f x, g x)

theorem StarAlgHom.coe_prod {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] [Semiring C] [Algebra R C] [Star C] (f : A →⋆ₐ[R] B) (g : A →⋆ₐ[R] C) : ⇑(f.prod g) = Pi.prod ⇑f ⇑g

@[simp]

theorem StarAlgHom.fst_prod {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] [Semiring C] [Algebra R C] [Star C] (f : A →⋆ₐ[R] B) (g : A →⋆ₐ[R] C) : (fst R B C).comp (f.prod g) = f

@[simp]

theorem StarAlgHom.snd_prod {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] [Semiring C] [Algebra R C] [Star C] (f : A →⋆ₐ[R] B) (g : A →⋆ₐ[R] C) : (snd R B C).comp (f.prod g) = g

@[simp]

theorem StarAlgHom.prod_fst_snd {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] : (fst R A B).prod (snd R A B) = 1

def StarAlgHom.prodEquiv {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] [Semiring C] [Algebra R C] [Star C] : (A →⋆ₐ[R] B) × (A →⋆ₐ[R] C) ≃ (A →⋆ₐ[R] B × C)

Taking the product of two maps with the same domain is equivalent to taking the product of their codomains.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem StarAlgHom.prodEquiv_apply {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] [Semiring C] [Algebra R C] [Star C] (f : (A →⋆ₐ[R] B) × (A →⋆ₐ[R] C)) : prodEquiv f = f.1.prod f.2

@[simp]

theorem StarAlgHom.prodEquiv_symm_apply {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] [Semiring C] [Algebra R C] [Star C] (f : A →⋆ₐ[R] B × C) : prodEquiv.symm f = ((fst R B C).comp f, (snd R B C).comp f)

Star algebra equivalences

structure StarAlgEquiv (R : Type u_1) (A : Type u_2) (B : Type u_3) [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] extends A ≃+* B : Type (max u_2 u_3)

A ⋆-algebra equivalence is an equivalence preserving addition, multiplication, scalar multiplication and the star operation, which allows for considering both unital and non-unital equivalences with a single structure. Currently, AlgEquiv requires unital algebras, which is why this structure does not extend it.

• toFun : A → B
• invFun : B → A
• left_inv : Function.LeftInverse self.invFun self.toFun
• right_inv : Function.RightInverse self.invFun self.toFun
• map_mul' (x y : A) : self.toFun (x * y) = self.toFun x * self.toFun y
• map_add' (x y : A) : self.toFun (x + y) = self.toFun x + self.toFun y
• map_star' (a : A) : self.toFun (star a) = star (self.toFun a)

  By definition, a ⋆-algebra equivalence preserves the star operation.

• map_smul' (r : R) (a : A) : self.toFun (r • a) = r • self.toFun a

  By definition, a ⋆-algebra equivalence commutes with the action of scalars.

def «term_≃⋆ₐ_» : Lean.TrailingParserDescr

A ⋆-algebra equivalence is an equivalence preserving addition, multiplication, scalar multiplication and the star operation, which allows for considering both unital and non-unital equivalences with a single structure. Currently, AlgEquiv requires unital algebras, which is why this structure does not extend it.

Equations

• «term_≃⋆ₐ_» = Lean.ParserDescr.trailingNode `«term_≃⋆ₐ_» 25 26 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " ≃⋆ₐ ") (Lean.ParserDescr.cat `term 25))

def «term_≃⋆ₐ[_]_» : Lean.TrailingParserDescr

A ⋆-algebra equivalence is an equivalence preserving addition, multiplication, scalar multiplication and the star operation, which allows for considering both unital and non-unital equivalences with a single structure. Currently, AlgEquiv requires unital algebras, which is why this structure does not extend it.

Equations

• One or more equations did not get rendered due to their size.

class NonUnitalAlgEquivClass (F : Type u_1) (R : outParam (Type u_2)) (A : outParam (Type u_3)) (B : outParam (Type u_4)) [Add A] [Mul A] [SMul R A] [Add B] [Mul B] [SMul R B] [EquivLike F A B] extends RingEquivClass F A B, MulActionSemiHomClass F id A B : Prop

The class that directly extends RingEquivClass and SMulHomClass.

Mostly an implementation detail for StarAlgEquivClass.

• map_mul (f : F) (a b : A) : f (a * b) = f a * f b
• map_add (f : F) (a b : A) : f (a + b) = f a + f b
• map_smulₛₗ (f : F) (c : R) (x : A) : f (c • x) = id c • f x

@[instance 100]

instance StarAlgEquivClass.instNonUnitalAlgHomClassOfNonUnitalAlgEquivClass {F : Type u_1} {R : Type u_2} {A : Type u_3} {B : Type u_4} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [EquivLike F A B] [NonUnitalAlgEquivClass F R A B] : NonUnitalAlgHomClass F R A B

@[instance 100]

instance StarAlgEquivClass.instAlgHomClass (F : Type u_1) (R : Type u_2) (A : Type u_3) (B : Type u_4) [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [EquivLike F A B] [NonUnitalAlgEquivClass F R A B] : AlgEquivClass F R A B

def StarAlgEquivClass.toStarAlgEquiv {F : Type u_1} {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Mul A] [SMul R A] [Star A] [Add B] [Mul B] [SMul R B] [Star B] [EquivLike F A B] [NonUnitalAlgEquivClass F R A B] [StarHomClass F A B] (f : F) : A ≃⋆ₐ[R] B

Turn an element of a type F satisfying AlgEquivClass F R A B and StarHomClass F A B into an actual StarAlgEquiv. This is declared as the default coercion from F to A ≃⋆ₐ[R] B.

Equations

• ↑f = { toRingEquiv := ↑f, map_star' := ⋯, map_smul' := ⋯ }

instance StarAlgEquivClass.instCoeHead {F : Type u_1} {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Mul A] [SMul R A] [Star A] [Add B] [Mul B] [SMul R B] [Star B] [EquivLike F A B] [NonUnitalAlgEquivClass F R A B] [StarHomClass F A B] : CoeHead F (A ≃⋆ₐ[R] B)

Any type satisfying AlgEquivClass and StarHomClass can be cast into StarAlgEquiv via StarAlgEquivClass.toStarAlgEquiv.

Equations

• StarAlgEquivClass.instCoeHead = { coe := StarAlgEquivClass.toStarAlgEquiv }

instance StarAlgEquiv.instEquivLike {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] : EquivLike (A ≃⋆ₐ[R] B) A B

Equations

• StarAlgEquiv.instEquivLike = { coe := fun (f : A ≃⋆ₐ[R] B) => f.toFun, inv := fun (f : A ≃⋆ₐ[R] B) => f.invFun, left_inv := ⋯, right_inv := ⋯, coe_injective' := ⋯ }

instance StarAlgEquiv.instNonUnitalAlgEquivClass {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] : NonUnitalAlgEquivClass (A ≃⋆ₐ[R] B) R A B

instance StarAlgEquiv.instStarHomClass {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] : StarHomClass (A ≃⋆ₐ[R] B) A B

instance StarAlgEquiv.instFunLike {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] : FunLike (A ≃⋆ₐ[R] B) A B

Helper instance for cases where the inference via EquivLike is too hard.

Equations

• StarAlgEquiv.instFunLike = { coe := fun (f : A ≃⋆ₐ[R] B) => f.toFun, coe_injective' := ⋯ }

@[simp]

theorem StarAlgEquiv.toRingEquiv_eq_coe {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (e : A ≃⋆ₐ[R] B) : e.toRingEquiv = ↑e

theorem StarAlgEquiv.ext {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] {f g : A ≃⋆ₐ[R] B} (h : ∀ (a : A), f a = g a) : f = g

theorem StarAlgEquiv.ext_iff {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] {f g : A ≃⋆ₐ[R] B} : f = g ↔ ∀ (a : A), f a = g a

def StarAlgEquiv.refl {R : Type u_2} {A : Type u_3} [Add A] [Mul A] [SMul R A] [Star A] : A ≃⋆ₐ[R] A

The identity map is a star algebra isomorphism.

Equations

• StarAlgEquiv.refl = { toRingEquiv := RingEquiv.refl A, map_star' := ⋯, map_smul' := ⋯ }

instance StarAlgEquiv.instInhabited {R : Type u_2} {A : Type u_3} [Add A] [Mul A] [SMul R A] [Star A] : Inhabited (A ≃⋆ₐ[R] A)

Equations

• StarAlgEquiv.instInhabited = { default := StarAlgEquiv.refl }

@[simp]

theorem StarAlgEquiv.coe_refl {R : Type u_2} {A : Type u_3} [Add A] [Mul A] [SMul R A] [Star A] : ⇑refl = id

def StarAlgEquiv.symm {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (e : A ≃⋆ₐ[R] B) : B ≃⋆ₐ[R] A

The inverse of a star algebra isomorphism is a star algebra isomorphism.

Equations

• e.symm = { toRingEquiv := e.symm, map_star' := ⋯, map_smul' := ⋯ }

def StarAlgEquiv.Simps.apply {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (e : A ≃⋆ₐ[R] B) : A → B

See Note [custom simps projection]

Equations

• StarAlgEquiv.Simps.apply e = ⇑e

def StarAlgEquiv.Simps.symm_apply {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (e : A ≃⋆ₐ[R] B) : B → A

See Note [custom simps projection]

Equations

• StarAlgEquiv.Simps.symm_apply e = ⇑e.symm

@[simp]

theorem StarAlgEquiv.invFun_eq_symm {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] {e : A ≃⋆ₐ[R] B} : EquivLike.inv e = ⇑e.symm

@[simp]

theorem StarAlgEquiv.symm_symm {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (e : A ≃⋆ₐ[R] B) : e.symm.symm = e

theorem StarAlgEquiv.symm_bijective {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] : Function.Bijective symm

@[simp]

theorem StarAlgEquiv.coe_mk {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (e : A ≃+* B) (h₁ : ∀ (a : A), e.toFun (star a) = star (e.toFun a)) (h₂ : ∀ (r : R) (a : A), e.toFun (r • a) = r • e.toFun a) : ⇑{ toRingEquiv := e, map_star' := h₁, map_smul' := h₂ } = ⇑e

@[simp]

theorem StarAlgEquiv.mk_coe {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (e : A ≃⋆ₐ[R] B) (e' : B → A) (h₁ : Function.LeftInverse e' ⇑e) (h₂ : Function.RightInverse e' ⇑e) (h₃ : ∀ (x y : A), { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun (x * y) = { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun x * { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun y) (h₄ : ∀ (x y : A), { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun (x + y) = { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun x + { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun y) (h₅ : ∀ (a : A), { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄ }.toFun (star a) = star ({ toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄ }.toFun a)) (h₆ : ∀ (r : R) (a : A), { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄ }.toFun (r • a) = r • { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄ }.toFun a) : { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄, map_star' := h₅, map_smul' := h₆ } = e

def StarAlgEquiv.symm_mk.aux {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (f : A → B) (f' : B → A) (h₁ : Function.LeftInverse f' f) (h₂ : Function.RightInverse f' f) (h₃ : ∀ (x y : A), { toFun := f, invFun := f', left_inv := h₁, right_inv := h₂ }.toFun (x * y) = { toFun := f, invFun := f', left_inv := h₁, right_inv := h₂ }.toFun x * { toFun := f, invFun := f', left_inv := h₁, right_inv := h₂ }.toFun y) (h₄ : ∀ (x y : A), { toFun := f, invFun := f', left_inv := h₁, right_inv := h₂ }.toFun (x + y) = { toFun := f, invFun := f', left_inv := h₁, right_inv := h₂ }.toFun x + { toFun := f, invFun := f', left_inv := h₁, right_inv := h₂ }.toFun y) (h₅ : ∀ (a : A), { toFun := f, invFun := f', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄ }.toFun (star a) = star ({ toFun := f, invFun := f', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄ }.toFun a)) (h₆ : ∀ (r : R) (a : A), { toFun := f, invFun := f', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄ }.toFun (r • a) = r • { toFun := f, invFun := f', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄ }.toFun a) : B ≃⋆ₐ[R] A

Auxiliary definition to avoid looping in dsimp with StarAlgEquiv.symm_mk.

Equations

• StarAlgEquiv.symm_mk.aux f f' h₁ h₂ h₃ h₄ h₅ h₆ = { toFun := f, invFun := f', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄, map_star' := h₅, map_smul' := h₆ }.symm

@[simp]

theorem StarAlgEquiv.symm_mk {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (f : A → B) (f' : B → A) (h₁ : Function.LeftInverse f' f) (h₂ : Function.RightInverse f' f) (h₃ : ∀ (x y : A), { toFun := f, invFun := f', left_inv := h₁, right_inv := h₂ }.toFun (x * y) = { toFun := f, invFun := f', left_inv := h₁, right_inv := h₂ }.toFun x * { toFun := f, invFun := f', left_inv := h₁, right_inv := h₂ }.toFun y) (h₄ : ∀ (x y : A), { toFun := f, invFun := f', left_inv := h₁, right_inv := h₂ }.toFun (x + y) = { toFun := f, invFun := f', left_inv := h₁, right_inv := h₂ }.toFun x + { toFun := f, invFun := f', left_inv := h₁, right_inv := h₂ }.toFun y) (h₅ : ∀ (a : A), { toFun := f, invFun := f', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄ }.toFun (star a) = star ({ toFun := f, invFun := f', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄ }.toFun a)) (h₆ : ∀ (r : R) (a : A), { toFun := f, invFun := f', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄ }.toFun (r • a) = r • { toFun := f, invFun := f', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄ }.toFun a) : { toFun := f, invFun := f', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄, map_star' := h₅, map_smul' := h₆ }.symm = let __src := symm_mk.aux f f' h₁ h₂ h₃ h₄ h₅ h₆; { toFun := f', invFun := f, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯, map_add' := ⋯, map_star' := ⋯, map_smul' := ⋯ }

@[simp]

theorem StarAlgEquiv.refl_symm {R : Type u_2} {A : Type u_3} [Add A] [Mul A] [SMul R A] [Star A] : refl.symm = refl

theorem StarAlgEquiv.to_ringEquiv_symm {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (f : A ≃⋆ₐ[R] B) : (↑f).symm = ↑f.symm

@[simp]

theorem StarAlgEquiv.symm_to_ringEquiv {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (e : A ≃⋆ₐ[R] B) : ↑e.symm = (↑e).symm

def StarAlgEquiv.trans {R : Type u_2} {A : Type u_3} {B : Type u_4} {C : Type u_5} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] [Add C] [Mul C] [SMul R C] [Star C] (e₁ : A ≃⋆ₐ[R] B) (e₂ : B ≃⋆ₐ[R] C) : A ≃⋆ₐ[R] C

Transitivity of StarAlgEquiv.

Equations

• e₁.trans e₂ = { toRingEquiv := e₁.trans e₂.toRingEquiv, map_star' := ⋯, map_smul' := ⋯ }

@[simp]

theorem StarAlgEquiv.apply_symm_apply {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (e : A ≃⋆ₐ[R] B) (x : B) : e (e.symm x) = x

@[simp]

theorem StarAlgEquiv.symm_apply_apply {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (e : A ≃⋆ₐ[R] B) (x : A) : e.symm (e x) = x

@[simp]

theorem StarAlgEquiv.symm_trans_apply {R : Type u_2} {A : Type u_3} {B : Type u_4} {C : Type u_5} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] [Add C] [Mul C] [SMul R C] [Star C] (e₁ : A ≃⋆ₐ[R] B) (e₂ : B ≃⋆ₐ[R] C) (x : C) : (e₁.trans e₂).symm x = e₁.symm (e₂.symm x)

@[simp]

theorem StarAlgEquiv.coe_trans {R : Type u_2} {A : Type u_3} {B : Type u_4} {C : Type u_5} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] [Add C] [Mul C] [SMul R C] [Star C] (e₁ : A ≃⋆ₐ[R] B) (e₂ : B ≃⋆ₐ[R] C) : ⇑(e₁.trans e₂) = ⇑e₂ ∘ ⇑e₁

@[simp]

theorem StarAlgEquiv.trans_apply {R : Type u_2} {A : Type u_3} {B : Type u_4} {C : Type u_5} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] [Add C] [Mul C] [SMul R C] [Star C] (e₁ : A ≃⋆ₐ[R] B) (e₂ : B ≃⋆ₐ[R] C) (x : A) : (e₁.trans e₂) x = e₂ (e₁ x)

theorem StarAlgEquiv.leftInverse_symm {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (e : A ≃⋆ₐ[R] B) : Function.LeftInverse ⇑e.symm ⇑e

theorem StarAlgEquiv.rightInverse_symm {R : Type u_2} {A : Type u_3} {B : Type u_4} [Add A] [Add B] [Mul A] [Mul B] [SMul R A] [SMul R B] [Star A] [Star B] (e : A ≃⋆ₐ[R] B) : Function.RightInverse ⇑e.symm ⇑e

def StarAlgEquiv.ofStarAlgHom {F : Type u_1} {G : Type u_2} {R : Type u_3} {A : Type u_4} {B : Type u_5} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [StarHomClass F A B] [FunLike G B A] (f : F) (g : G) (h₁ : ∀ (x : A), g (f x) = x) (h₂ : ∀ (x : B), f (g x) = x) : A ≃⋆ₐ[R] B

If a (unital or non-unital) star algebra morphism has an inverse, it is an isomorphism of star algebras.

Equations

• StarAlgEquiv.ofStarAlgHom f g h₁ h₂ = { toFun := ⇑f, invFun := ⇑g, left_inv := h₁, right_inv := h₂, map_mul' := ⋯, map_add' := ⋯, map_star' := ⋯, map_smul' := ⋯ }

@[simp]

theorem StarAlgEquiv.ofStarAlgHom_symm_apply {F : Type u_1} {G : Type u_2} {R : Type u_3} {A : Type u_4} {B : Type u_5} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [StarHomClass F A B] [FunLike G B A] (f : F) (g : G) (h₁ : ∀ (x : A), g (f x) = x) (h₂ : ∀ (x : B), f (g x) = x) (a : B) : (ofStarAlgHom f g h₁ h₂).symm a = g a

@[simp]

theorem StarAlgEquiv.ofStarAlgHom_apply {F : Type u_1} {G : Type u_2} {R : Type u_3} {A : Type u_4} {B : Type u_5} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [StarHomClass F A B] [FunLike G B A] (f : F) (g : G) (h₁ : ∀ (x : A), g (f x) = x) (h₂ : ∀ (x : B), f (g x) = x) (a : A) : (ofStarAlgHom f g h₁ h₂) a = f a

noncomputable def StarAlgEquiv.ofBijective {F : Type u_1} {R : Type u_3} {A : Type u_4} {B : Type u_5} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [StarHomClass F A B] (f : F) (hf : Function.Bijective ⇑f) : A ≃⋆ₐ[R] B

Promote a bijective star algebra homomorphism to a star algebra equivalence.

Equations

• StarAlgEquiv.ofBijective f hf = { toFun := ⇑f, invFun := (RingEquiv.ofBijective f hf).invFun, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯, map_add' := ⋯, map_star' := ⋯, map_smul' := ⋯ }

@[simp]

theorem StarAlgEquiv.coe_ofBijective {F : Type u_1} {R : Type u_3} {A : Type u_4} {B : Type u_5} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [StarHomClass F A B] {f : F} (hf : Function.Bijective ⇑f) : ⇑(ofBijective f hf) = ⇑f

theorem StarAlgEquiv.ofBijective_apply {F : Type u_1} {R : Type u_3} {A : Type u_4} {B : Type u_5} [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [StarHomClass F A B] {f : F} (hf : Function.Bijective ⇑f) (a : A) : (ofBijective f hf) a = f a