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Mathlib.Analysis.NormedSpace.Star.Basic

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⋆ * x‖ = ‖x‖^2 for all x. 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 #

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

Instances

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

    Instances For

      The star map in a normed star group is an isometry

      A C*-ring is a normed star ring that satisfies the stronger condition ‖x⋆ * x‖ = ‖x‖^2 for every x.

      Instances

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

        @[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
        @[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
        instance Pi.starRing' {ι : Type u_4} {R : ιType 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.

        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)
        instance Pi.cstarRing {ι : Type u_4} {R : ιType u_7} [(i : ι) → NonUnitalNormedRing (R i)] [(i : ι) → StarRing (R i)] [Fintype ι] [∀ (i : ι), CstarRing (R i)] :
        CstarRing ((i : ι) → R i)
        instance Pi.cstarRing' {ι : Type u_4} {R₁ : Type u_5} [NonUnitalNormedRing R₁] [StarRing R₁] [CstarRing R₁] [Fintype ι] :
        CstarRing (ιR₁)
        @[simp]
        theorem CstarRing.norm_one {E : Type u_2} [NormedRing E] [StarRing E] [CstarRing E] [Nontrivial E] :
        theorem CstarRing.norm_coe_unitary {E : Type u_2} [NormedRing E] [StarRing E] [CstarRing E] [Nontrivial E] (U : { x // x 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) :
        @[simp]
        theorem CstarRing.norm_coe_unitary_mul {E : Type u_2} [NormedRing E] [StarRing E] [CstarRing E] (U : { x // x unitary E }) (A : E) :
        U * A = A
        @[simp]
        theorem CstarRing.norm_unitary_smul {E : Type u_2} [NormedRing E] [StarRing E] [CstarRing E] (U : { x // x unitary E }) (A : E) :
        theorem CstarRing.norm_mem_unitary_mul {E : Type u_2} [NormedRing E] [StarRing E] [CstarRing E] {U : E} (A : E) (hU : U unitary E) :
        @[simp]
        theorem CstarRing.norm_mul_coe_unitary {E : Type u_2} [NormedRing E] [StarRing E] [CstarRing E] (A : E) (U : { x // x 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) :
        theorem IsSelfAdjoint.nnnorm_pow_two_pow {E : Type u_2} [NormedRing E] [StarRing E] [CstarRing E] {x : E} (hx : IsSelfAdjoint x) (n : ) :
        x ^ 2 ^ n‖₊ = x‖₊ ^ 2 ^ n
        theorem selfAdjoint.nnnorm_pow_two_pow {E : Type u_2} [NormedRing E] [StarRing E] [CstarRing E] (x : { x // x selfAdjoint E }) (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] :

        star bundled as a linear isometric equivalence

        Instances For
          @[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
          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 { x // x S }
          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 { x // x S }