Unitary elements of a star monoid

This file defines unitary R, where R is a star monoid, as the submonoid made of the elements that satisfy star U * U = 1 and U * star U = 1, and these form a group. This includes, for instance, unitary operators on Hilbert spaces.

See also Matrix.UnitaryGroup for specializations to unitary (Matrix n n R).

Tags

unitary

def unitary (R : Type u_1) [Monoid R] [StarMul R] : Submonoid R

In a *-monoid, unitary R is the submonoid consisting of all the elements U of R such that star U * U = 1 and U * star U = 1.

Equations

• unitary R = { carrier := {U : R | star U * U = 1 ∧ U * star U = 1}, mul_mem' := ⋯, one_mem' := ⋯ }

theorem unitary.mem_iff {R : Type u_1} [Monoid R] [StarMul R] {U : R} : U ∈ unitary R ↔ star U * U = 1 ∧ U * star U = 1

@[simp]

theorem unitary.star_mul_self_of_mem {R : Type u_1} [Monoid R] [StarMul R] {U : R} (hU : U ∈ unitary R) : star U * U = 1

@[simp]

theorem unitary.mul_star_self_of_mem {R : Type u_1} [Monoid R] [StarMul R] {U : R} (hU : U ∈ unitary R) : U * star U = 1

theorem unitary.star_mem {R : Type u_1} [Monoid R] [StarMul R] {U : R} (hU : U ∈ unitary R) : star U ∈ unitary R

@[simp]

theorem unitary.star_mem_iff {R : Type u_1} [Monoid R] [StarMul R] {U : R} : star U ∈ unitary R ↔ U ∈ unitary R

instance unitary.instStarSubtypeMemSubmonoid {R : Type u_1} [Monoid R] [StarMul R] : Star ↥(unitary R)

Equations

• unitary.instStarSubtypeMemSubmonoid = { star := fun (U : ↥(unitary R)) => ⟨star ↑U, ⋯⟩ }

@[simp]

theorem unitary.coe_star {R : Type u_1} [Monoid R] [StarMul R] {U : ↥(unitary R)} : ↑(star U) = star ↑U

theorem unitary.coe_star_mul_self {R : Type u_1} [Monoid R] [StarMul R] (U : ↥(unitary R)) : star ↑U * ↑U = 1

theorem unitary.coe_mul_star_self {R : Type u_1} [Monoid R] [StarMul R] (U : ↥(unitary R)) : ↑U * ↑(star U) = 1

@[simp]

theorem unitary.star_mul_self {R : Type u_1} [Monoid R] [StarMul R] (U : ↥(unitary R)) : star U * U = 1

@[simp]

theorem unitary.mul_star_self {R : Type u_1} [Monoid R] [StarMul R] (U : ↥(unitary R)) : U * star U = 1

instance unitary.instGroupSubtypeMemSubmonoid {R : Type u_1} [Monoid R] [StarMul R] : Group ↥(unitary R)

Equations

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

instance unitary.instInvolutiveStarSubtypeMemSubmonoid {R : Type u_1} [Monoid R] [StarMul R] : InvolutiveStar ↥(unitary R)

Equations

• unitary.instInvolutiveStarSubtypeMemSubmonoid = { toStar := unitary.instStarSubtypeMemSubmonoid, star_involutive := ⋯ }

instance unitary.instStarMulSubtypeMemSubmonoid {R : Type u_1} [Monoid R] [StarMul R] : StarMul ↥(unitary R)

Equations

• unitary.instStarMulSubtypeMemSubmonoid = { toInvolutiveStar := unitary.instInvolutiveStarSubtypeMemSubmonoid, star_mul := ⋯ }

instance unitary.instInhabitedSubtypeMemSubmonoid {R : Type u_1} [Monoid R] [StarMul R] : Inhabited ↥(unitary R)

Equations

• unitary.instInhabitedSubtypeMemSubmonoid = { default := 1 }

theorem unitary.star_eq_inv {R : Type u_1} [Monoid R] [StarMul R] (U : ↥(unitary R)) : star U = U⁻¹

theorem unitary.star_eq_inv' {R : Type u_1} [Monoid R] [StarMul R] : star = Inv.inv

def unitary.toUnits {R : Type u_1} [Monoid R] [StarMul R] : ↥(unitary R) →* Rˣ

The unitary elements embed into the units.

Equations

• unitary.toUnits = { toFun := fun (x : ↥(unitary R)) => { val := ↑x, inv := ↑x⁻¹, val_inv := ⋯, inv_val := ⋯ }, map_one' := ⋯, map_mul' := ⋯ }

@[simp]

theorem unitary.val_toUnits_apply {R : Type u_1} [Monoid R] [StarMul R] (x : ↥(unitary R)) : ↑(toUnits x) = ↑x

@[simp]

theorem unitary.val_inv_toUnits_apply {R : Type u_1} [Monoid R] [StarMul R] (x : ↥(unitary R)) : ↑(toUnits x)⁻¹ = ↑x⁻¹

theorem unitary.toUnits_injective {R : Type u_1} [Monoid R] [StarMul R] : Function.Injective ⇑toUnits

theorem IsUnit.mem_unitary_iff_star_mul_self {R : Type u_1} [Monoid R] [StarMul R] {u : R} (hu : IsUnit u) : u ∈ unitary R ↔ star u * u = 1

theorem IsUnit.mem_unitary_iff_mul_star_self {R : Type u_1} [Monoid R] [StarMul R] {u : R} (hu : IsUnit u) : u ∈ unitary R ↔ u * star u = 1

theorem IsUnit.mem_unitary_of_star_mul_self {R : Type u_1} [Monoid R] [StarMul R] {u : R} (hu : IsUnit u) : star u * u = 1 → u ∈ unitary R

Alias of the reverse direction of IsUnit.mem_unitary_iff_star_mul_self.

theorem IsUnit.mem_unitary_of_mul_star_self {R : Type u_1} [Monoid R] [StarMul R] {u : R} (hu : IsUnit u) : u * star u = 1 → u ∈ unitary R

Alias of the reverse direction of IsUnit.mem_unitary_iff_mul_star_self.

theorem unitary.mul_left_inj {R : Type u_1} [Monoid R] [StarMul R] {x y : R} (U : ↥(unitary R)) : x * ↑U = y * ↑U ↔ x = y

For unitary U in a star-monoid R, x * U = y * U if and only if x = y for all x and y in R.

theorem unitary.mul_right_inj {R : Type u_1} [Monoid R] [StarMul R] {x y : R} (U : ↥(unitary R)) : ↑U * x = ↑U * y ↔ x = y

For unitary U in a star-monoid R, U * x = U * y if and only if x = y for all x and y in R.

theorem unitary.mul_inv_mem_iff {G : Type u_2} [Group G] [StarMul G] (a b : G) : a * b⁻¹ ∈ unitary G ↔ star a * a = star b * b

theorem unitary.inv_mul_mem_iff {G : Type u_2} [Group G] [StarMul G] (a b : G) : a⁻¹ * b ∈ unitary G ↔ a * star a = b * star b

theorem Units.unitary_eq {R : Type u_1} [Monoid R] [StarMul R] : unitary Rˣ = Submonoid.comap (coeHom R) (unitary R)

theorem Units.mul_inv_mem_unitary {R : Type u_1} [Monoid R] [StarMul R] (a b : Rˣ) : ↑a * ↑b⁻¹ ∈ unitary R ↔ star a * a = star b * b

In a star monoid, the product a * b⁻¹ of units is unitary if star a * a = star b * b.

theorem Units.inv_mul_mem_unitary {R : Type u_1} [Monoid R] [StarMul R] (a b : Rˣ) : ↑a⁻¹ * ↑b ∈ unitary R ↔ a * star a = b * star b

In a star monoid, the product a⁻¹ * b of units is unitary if a * star a = b * star b.

instance unitary.instIsStarNormal {R : Type u_1} [Monoid R] [StarMul R] (u : ↥(unitary R)) : IsStarNormal u

instance unitary.coe_isStarNormal {R : Type u_1} [Monoid R] [StarMul R] (u : ↥(unitary R)) : IsStarNormal ↑u

theorem isStarNormal_of_mem_unitary {R : Type u_1} [Monoid R] [StarMul R] {u : R} (hu : u ∈ unitary R) : IsStarNormal u

theorem unitary.inv_mem {G : Type u_2} [Group G] [StarMul G] {g : G} (hg : g ∈ unitary G) : g⁻¹ ∈ unitary G

def unitarySubgroup (G : Type u_2) [Group G] [StarMul G] : Subgroup G

unitary as a Subgroup of a group.

Note the group structure on this type is not defeq to the one on unitary. This situation naturally arises when considering the unitary elements as a subgroup of the group of units of a star monoid.

Equations

• unitarySubgroup G = { toSubmonoid := unitary G, inv_mem' := ⋯ }

@[simp]

theorem unitarySubgroup_toSubmonoid {G : Type u_2} [Group G] [StarMul G] : (unitarySubgroup G).toSubmonoid = unitary G

@[simp]

theorem mem_unitarySubgroup_iff {G : Type u_2} [Group G] [StarMul G] {g : G} : g ∈ unitarySubgroup G ↔ g ∈ unitary G

theorem unitary.inv_mem_iff {G : Type u_2} [Group G] [StarMul G] {g : G} : g⁻¹ ∈ unitary G ↔ g ∈ unitary G

theorem unitary.smul_mem_of_mem {R : Type u_1} {A : Type u_2} [Monoid R] [Monoid A] [MulAction R A] [SMulCommClass R A A] [IsScalarTower R A A] [StarMul R] [StarMul A] [StarModule R A] {r : R} {a : A} (hr : r ∈ unitary R) (ha : a ∈ unitary A) : r • a ∈ unitary A

theorem unitary.smul_mem {R : Type u_1} {A : Type u_2} [Monoid R] [Monoid A] [MulAction R A] [SMulCommClass R A A] [IsScalarTower R A A] [StarMul R] [StarMul A] [StarModule R A] (r : ↥(unitary R)) {a : A} (ha : a ∈ unitary A) : r • a ∈ unitary A

instance unitary.instSMulSubtypeMemSubmonoid {R : Type u_1} {A : Type u_2} [Monoid R] [Monoid A] [MulAction R A] [SMulCommClass R A A] [IsScalarTower R A A] [StarMul R] [StarMul A] [StarModule R A] : SMul ↥(unitary R) ↥(unitary A)

Equations

• unitary.instSMulSubtypeMemSubmonoid = { smul := fun (r : ↥(unitary R)) (a : ↥(unitary A)) => ⟨r • ↑a, ⋯⟩ }

@[simp]

theorem unitary.coe_smul {R : Type u_1} {A : Type u_2} [Monoid R] [Monoid A] [MulAction R A] [SMulCommClass R A A] [IsScalarTower R A A] [StarMul R] [StarMul A] [StarModule R A] (r : ↥(unitary R)) (a : ↥(unitary A)) : ↑(r • a) = r • ↑a

instance unitary.instMulActionSubtypeMemSubmonoid {R : Type u_1} {A : Type u_2} [Monoid R] [Monoid A] [MulAction R A] [SMulCommClass R A A] [IsScalarTower R A A] [StarMul R] [StarMul A] [StarModule R A] : MulAction ↥(unitary R) ↥(unitary A)

Equations

• unitary.instMulActionSubtypeMemSubmonoid = { toSMul := unitary.instSMulSubtypeMemSubmonoid, one_smul := ⋯, mul_smul := ⋯ }

instance unitary.instStarModuleSubtypeMemSubmonoid {R : Type u_1} {A : Type u_2} [Monoid R] [Monoid A] [MulAction R A] [SMulCommClass R A A] [IsScalarTower R A A] [StarMul R] [StarMul A] [StarModule R A] : StarModule ↥(unitary R) ↥(unitary A)

instance unitary.instSMulCommClassSubtypeMemSubmonoid {R : Type u_1} {S : Type u_2} {A : Type u_3} [Monoid R] [Monoid S] [Monoid A] [StarMul R] [StarMul S] [StarMul A] [MulAction R A] [MulAction S A] [StarModule R A] [StarModule S A] [IsScalarTower R A A] [IsScalarTower S A A] [SMulCommClass R A A] [SMulCommClass S A A] [SMulCommClass R S A] : SMulCommClass ↥(unitary R) ↥(unitary S) ↥(unitary A)

instance unitary.instIsScalarTowerSubtypeMemSubmonoid {R : Type u_1} {S : Type u_2} {A : Type u_3} [Monoid R] [Monoid S] [Monoid A] [StarMul R] [StarMul S] [StarMul A] [MulAction R S] [MulAction R A] [MulAction S A] [StarModule R S] [StarModule R A] [StarModule S A] [IsScalarTower R A A] [IsScalarTower S A A] [SMulCommClass R A A] [SMulCommClass S A A] [IsScalarTower R S S] [SMulCommClass R S S] [IsScalarTower R S A] : IsScalarTower ↥(unitary R) ↥(unitary S) ↥(unitary A)

theorem unitary.map_mem {R : Type u_2} {S : Type u_3} [Monoid R] [StarMul R] [Monoid S] [StarMul S] {F : Type u_5} [FunLike F R S] [StarHomClass F R S] [MonoidHomClass F R S] (f : F) {r : R} (hr : r ∈ unitary R) : f r ∈ unitary S

def unitary.map {R : Type u_2} {S : Type u_3} [Monoid R] [StarMul R] [Monoid S] [StarMul S] (f : R →⋆* S) : ↥(unitary R) →⋆* ↥(unitary S)

The star monoid homomorphism between unitary subgroups induced by a star monoid homomorphism of the underlying star monoids.

Equations

• unitary.map f = { toFun := Subtype.map ⇑f ⋯, map_one' := ⋯, map_mul' := ⋯, map_star' := ⋯ }

@[simp]

theorem unitary.map_coe {R : Type u_2} {S : Type u_3} [Monoid R] [StarMul R] [Monoid S] [StarMul S] (f : R →⋆* S) (a✝ : ↥(unitary R)) : (map f) a✝ = Subtype.map ⇑f ⋯ a✝

@[simp]

theorem unitary.coe_map {R : Type u_2} {S : Type u_3} [Monoid R] [StarMul R] [Monoid S] [StarMul S] (f : R →⋆* S) (x : ↥(unitary R)) : ↑((map f) x) = f ↑x

@[simp]

theorem unitary.coe_map_star {R : Type u_2} {S : Type u_3} [Monoid R] [StarMul R] [Monoid S] [StarMul S] (f : R →⋆* S) (x : ↥(unitary R)) : ↑((map f) (star x)) = f (star ↑x)

@[simp]

theorem unitary.map_id {R : Type u_2} [Monoid R] [StarMul R] : map (StarMonoidHom.id R) = StarMonoidHom.id ↥(unitary R)

theorem unitary.map_comp {R : Type u_2} {S : Type u_3} {T : Type u_4} [Monoid R] [StarMul R] [Monoid S] [StarMul S] [Monoid T] [StarMul T] (g : S →⋆* T) (f : R →⋆* S) : map (g.comp f) = (map g).comp (map f)

@[simp]

theorem unitary.map_injective {R : Type u_2} {S : Type u_3} [Monoid R] [StarMul R] [Monoid S] [StarMul S] {f : R →⋆* S} (hf : Function.Injective ⇑f) : Function.Injective ⇑(map f)

theorem unitary.toUnits_comp_map {R : Type u_2} {S : Type u_3} [Monoid R] [StarMul R] [Monoid S] [StarMul S] (f : R →⋆* S) : toUnits.comp (map f).toMonoidHom = (Units.map f.toMonoidHom).comp toUnits

def unitary.mapEquiv {R : Type u_2} {S : Type u_3} [Monoid R] [StarMul R] [Monoid S] [StarMul S] (f : R ≃⋆* S) : ↥(unitary R) ≃⋆* ↥(unitary S)

The star monoid isomorphism between unitary subgroups induced by a star monoid isomorphism of the underlying star monoids.

Equations

• unitary.mapEquiv f = { toFun := ⇑(unitary.map f.toStarMonoidHom), invFun := ⇑(unitary.map f.symm.toStarMonoidHom), left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯, map_star' := ⋯ }

@[simp]

theorem unitary.mapEquiv_apply {R : Type u_2} {S : Type u_3} [Monoid R] [StarMul R] [Monoid S] [StarMul S] (f : R ≃⋆* S) (a : ↥(unitary R)) : (mapEquiv f) a = (map f.toStarMonoidHom) a

@[simp]

theorem unitary.mapEquiv_symm_apply {R : Type u_2} {S : Type u_3} [Monoid R] [StarMul R] [Monoid S] [StarMul S] (f : R ≃⋆* S) (a : ↥(unitary S)) : (mapEquiv f).symm a = (map f.symm.toStarMonoidHom) a

@[simp]

theorem unitary.mapEquiv_refl {R : Type u_2} [Monoid R] [StarMul R] : mapEquiv (StarMulEquiv.refl R) = StarMulEquiv.refl ↥(unitary R)

@[simp]

theorem unitary.mapEquiv_symm {R : Type u_2} {S : Type u_3} [Monoid R] [StarMul R] [Monoid S] [StarMul S] (f : R ≃⋆* S) : mapEquiv f.symm = (mapEquiv f).symm

@[simp]

theorem unitary.mapEquiv_trans {R : Type u_2} {S : Type u_3} {T : Type u_4} [Monoid R] [StarMul R] [Monoid S] [StarMul S] [Monoid T] [StarMul T] (f : R ≃⋆* S) (g : S ≃⋆* T) : mapEquiv (f.trans g) = (mapEquiv f).trans (mapEquiv g)

@[simp]

theorem unitary.toMonoidHom_mapEquiv {R : Type u_2} {S : Type u_3} [Monoid R] [StarMul R] [Monoid S] [StarMul S] (f : R ≃⋆* S) : (mapEquiv f).toStarMonoidHom = map f.toStarMonoidHom

def unitarySubgroupUnitsEquiv {M : Type u_5} [Monoid M] [StarMul M] : ↥(unitarySubgroup Mˣ) ≃* ↥(unitary M)

The unitary subgroup of the units is equivalent to the unitary elements of the monoid.

Equations

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

@[simp]

theorem val_inv_unitarySubgroupUnitsEquiv_symm_apply_coe {M : Type u_5} [Monoid M] [StarMul M] (x : ↥(unitary M)) : ↑(↑(unitarySubgroupUnitsEquiv.symm x))⁻¹ = star ↑x

@[simp]

theorem val_unitarySubgroupUnitsEquiv_symm_apply_coe {M : Type u_5} [Monoid M] [StarMul M] (x : ↥(unitary M)) : ↑↑(unitarySubgroupUnitsEquiv.symm x) = ↑x

@[simp]

theorem unitarySubgroupUnitsEquiv_apply_coe {M : Type u_5} [Monoid M] [StarMul M] (x : ↥(unitarySubgroup Mˣ)) : ↑(unitarySubgroupUnitsEquiv x) = ↑↑x

instance unitary.instCommGroupSubtypeMemSubmonoid {R : Type u_1} [CommMonoid R] [StarMul R] : CommGroup ↥(unitary R)

Equations

• unitary.instCommGroupSubtypeMemSubmonoid = { toGroup := inferInstanceAs (Group ↥(unitary R)), mul_comm := ⋯ }

theorem unitary.mem_iff_star_mul_self {R : Type u_1} [CommMonoid R] [StarMul R] {U : R} : U ∈ unitary R ↔ star U * U = 1

theorem unitary.mem_iff_self_mul_star {R : Type u_1} [CommMonoid R] [StarMul R] {U : R} : U ∈ unitary R ↔ U * star U = 1

theorem unitary.coe_inv {R : Type u_1} [GroupWithZero R] [StarMul R] (U : ↥(unitary R)) : ↑U⁻¹ = (↑U)⁻¹

theorem unitary.coe_div {R : Type u_1} [GroupWithZero R] [StarMul R] (U₁ U₂ : ↥(unitary R)) : ↑(U₁ / U₂) = ↑U₁ / ↑U₂

theorem unitary.coe_zpow {R : Type u_1} [GroupWithZero R] [StarMul R] (U : ↥(unitary R)) (z : ℤ) : ↑(U ^ z) = ↑U ^ z

instance unitary.instNegSubtypeMemSubmonoid {R : Type u_1} [Ring R] [StarRing R] : Neg ↥(unitary R)

Equations

• unitary.instNegSubtypeMemSubmonoid = { neg := fun (U : ↥(unitary R)) => ⟨-↑U, ⋯⟩ }

theorem unitary.coe_neg {R : Type u_1} [Ring R] [StarRing R] (U : ↥(unitary R)) : ↑(-U) = -↑U

instance unitary.instHasDistribNegSubtypeMemSubmonoid {R : Type u_1} [Ring R] [StarRing R] : HasDistribNeg ↥(unitary R)

Equations

• unitary.instHasDistribNegSubtypeMemSubmonoid = Function.Injective.hasDistribNeg (fun (a : ↥(unitary R)) => ↑a) ⋯ ⋯ ⋯

@[simp]

theorem unitary.spectrum.unitary_conjugate {R : Type u_2} {A : Type u_3} [CommSemiring R] [Ring A] [Algebra R A] [StarMul A] {a : A} {u : ↥(unitary A)} : spectrum R (↑u * a * star ↑u) = spectrum R a

Unitary conjugation preserves the spectrum, star on left.

@[simp]

theorem unitary.spectrum.unitary_conjugate' {R : Type u_2} {A : Type u_3} [CommSemiring R] [Ring A] [Algebra R A] [StarMul A] {a : A} {u : ↥(unitary A)} : spectrum R (star ↑u * a * ↑u) = spectrum R a

Unitary conjugation preserves the spectrum, star on right.

theorem IsStarProjection.two_mul_sub_one_mem_unitary {R : Type u_2} [Ring R] [StarRing R] {p : R} (hp : IsStarProjection p) : 2 * p - 1 ∈ unitary R