# mathlibdocumentation

analysis.normed_space.star

# Normed star rings and algebras #

A normed star monoid is a star_add_monoid endowed with a norm such that the star operation is isometric.

A C⋆-ring is a normed star monoid 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 [normed_field 𝕜] [star_ring 𝕜] [normed_ring E] [star_ring E] [cstar_ring E] [normed_algebra 𝕜 E] [star_module 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]
structure normed_star_monoid (E : Type u_1) [normed_group E]  :
Type

A normed star ring is a star ring endowed with a norm such that star is isometric.

Instances
@[class]
structure cstar_ring (E : Type u_1) [normed_ring E] [star_ring E] :
Type

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

Instances
@[protected, instance]
noncomputable def real.cstar_ring  :
Equations
def star_normed_group_hom {E : Type u_2} [normed_group E]  :

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

Equations
theorem star_isometry {E : Type u_2} [normed_group E]  :

The star map in a normed star group is an isometry

theorem continuous_star {E : Type u_2} [normed_group E]  :
theorem continuous_on_star {E : Type u_2} [normed_group E] {s : set E} :
theorem continuous_at_star {E : Type u_2} [normed_group E] {x : E} :
theorem continuous_within_at_star {E : Type u_2} [normed_group E] {s : set E} {x : E} :
theorem tendsto_star {E : Type u_2} [normed_group E] (x : E) :
(𝓝 x) (𝓝 (star x))
theorem filter.tendsto.star {E : Type u_2} {α : Type u_3} [normed_group E] {f : α → E} {l : filter α} {y : E} (h : (𝓝 y)) :
filter.tendsto (λ (x : α), star (f x)) l (𝓝 (star y))
theorem continuous.star {E : Type u_2} {α : Type u_3} [normed_group E] {f : α → E} (hf : continuous f) :
continuous (λ (y : α), star (f y))
theorem continuous_at.star {E : Type u_2} {α : Type u_3} [normed_group E] {f : α → E} {x : α} (hf : x) :
continuous_at (λ (x : α), star (f x)) x
theorem continuous_on.star {E : Type u_2} {α : Type u_3} [normed_group E] {f : α → E} {s : set α} (hf : s) :
continuous_on (λ (x : α), star (f x)) s
theorem continuous_within_at.star {E : Type u_2} {α : Type u_3} [normed_group E] {f : α → E} {s : set α} {x : α} (hf : x) :
continuous_within_at (λ (x : α), star (f x)) s x
@[protected, instance]
def ring_hom_isometric.star_ring_end {E : Type u_2} [star_ring E]  :
@[protected, instance]

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

Equations
theorem cstar_ring.norm_self_mul_star {E : Type u_2} [normed_ring E] [star_ring E] [cstar_ring E] {x : E} :
theorem cstar_ring.norm_star_mul_self' {E : Type u_2} [normed_ring E] [star_ring E] [cstar_ring E] {x : E} :
@[simp]
theorem cstar_ring.norm_one {E : Type u_2} [normed_ring E] [star_ring E] [cstar_ring E] [nontrivial E] :
@[protected, instance]
def cstar_ring.norm_one_class {E : Type u_2} [normed_ring E] [star_ring E] [cstar_ring E] [nontrivial E] :
theorem cstar_ring.norm_coe_unitary {E : Type u_2} [normed_ring E] [star_ring E] [cstar_ring E] [nontrivial E] (U : (unitary E)) :
@[simp]
theorem cstar_ring.norm_of_mem_unitary {E : Type u_2} [normed_ring E] [star_ring E] [cstar_ring E] [nontrivial E] {U : E} (hU : U ) :
@[simp]
theorem cstar_ring.norm_coe_unitary_mul {E : Type u_2} [normed_ring E] [star_ring E] [cstar_ring E] (U : (unitary E)) (A : E) :
@[simp]
theorem cstar_ring.norm_unitary_smul {E : Type u_2} [normed_ring E] [star_ring E] [cstar_ring E] (U : (unitary E)) (A : E) :
theorem cstar_ring.norm_mem_unitary_mul {E : Type u_2} [normed_ring E] [star_ring E] [cstar_ring E] {U : E} (A : E) (hU : U ) :
@[simp]
theorem cstar_ring.norm_mul_coe_unitary {E : Type u_2} [normed_ring E] [star_ring E] [cstar_ring E] (A : E) (U : (unitary E)) :
theorem cstar_ring.norm_mul_mem_unitary {E : Type u_2} [normed_ring E] [star_ring E] [cstar_ring E] (A : E) {U : E} (hU : U ) :
def starₗᵢ (𝕜 : Type u_1) {E : Type u_2} [star_ring 𝕜] [normed_ring E] [star_ring E] [ E] [ E] :

star bundled as a linear isometric equivalence

Equations
@[simp]
theorem coe_starₗᵢ {𝕜 : Type u_1} {E : Type u_2} [star_ring 𝕜] [normed_ring E] [star_ring E] [ E] [ E] :
theorem starₗᵢ_apply {𝕜 : Type u_1} {E : Type u_2} [star_ring 𝕜] [normed_ring E] [star_ring E] [ E] [ E] {x : E} :
(starₗᵢ 𝕜) x = star x