# Documentation

Mathlib.Topology.Algebra.StarSubalgebra

# Topological star (sub)algebras #

A topological star algebra over a topological semiring R is a topological semiring with a compatible continuous scalar multiplication by elements of R and a continuous star operation. We reuse typeclass ContinuousSMul for topological algebras.

## Results #

This is just a minimal stub for now!

The topological closure of a star subalgebra is still a star subalgebra, which as a star algebra is a topological star algebra.

theorem StarSubalgebra.embedding_inclusion {R : Type u_1} {A : Type u_2} [] [] [] [] [Algebra R A] [] [] {S₁ : } {S₂ : } (h : S₁ S₂) :

The StarSubalgebra.inclusion of a star subalgebra is an Embedding.

theorem StarSubalgebra.closedEmbedding_inclusion {R : Type u_1} {A : Type u_2} [] [] [] [] [Algebra R A] [] [] {S₁ : } {S₂ : } (h : S₁ S₂) (hS₁ : IsClosed S₁) :

The StarSubalgebra.inclusion of a closed star subalgebra is a ClosedEmbedding.

def StarSubalgebra.topologicalClosure {R : Type u_1} {A : Type u_2} [] [] [] [] [Algebra R A] [] [] [] (s : ) :

The closure of a star subalgebra in a topological star algebra as a star subalgebra.

Instances For
theorem StarSubalgebra.topologicalClosure_toSubalgebra_comm {R : Type u_1} {A : Type u_2} [] [] [] [] [Algebra R A] [] [] [] (s : ) :
().toSubalgebra = Subalgebra.topologicalClosure s.toSubalgebra
@[simp]
theorem StarSubalgebra.topologicalClosure_coe {R : Type u_1} {A : Type u_2} [] [] [] [] [Algebra R A] [] [] [] (s : ) :
theorem StarSubalgebra.le_topologicalClosure {R : Type u_1} {A : Type u_2} [] [] [] [] [Algebra R A] [] [] [] (s : ) :
theorem StarSubalgebra.isClosed_topologicalClosure {R : Type u_1} {A : Type u_2} [] [] [] [] [Algebra R A] [] [] [] (s : ) :
theorem StarSubalgebra.topologicalClosure_minimal {R : Type u_1} {A : Type u_2} [] [] [] [] [Algebra R A] [] [] [] {s : } {t : } (h : s t) (ht : IsClosed t) :
theorem StarSubalgebra.topologicalClosure_mono {R : Type u_1} {A : Type u_2} [] [] [] [] [Algebra R A] [] [] [] :
Monotone StarSubalgebra.topologicalClosure
theorem StarSubalgebra.topologicalClosure_map_le {R : Type u_1} {A : Type u_2} {B : Type u_3} [] [] [] [] [Algebra R A] [] [] [] [] [] [Algebra R B] [] [] [] (s : ) (φ : A →⋆ₐ[R] B) (hφ : ) :
theorem StarSubalgebra.map_topologicalClosure_le {R : Type u_1} {A : Type u_2} {B : Type u_3} [] [] [] [] [Algebra R A] [] [] [] [] [] [Algebra R B] [] [] [] (s : ) (φ : A →⋆ₐ[R] B) (hφ : ) :
theorem StarSubalgebra.topologicalClosure_map {R : Type u_1} {A : Type u_2} {B : Type u_3} [] [] [] [] [Algebra R A] [] [] [] [] [] [Algebra R B] [] [] [] (s : ) (φ : A →⋆ₐ[R] B) (hφ : ) :
theorem Subalgebra.topologicalClosure_star_comm {R : Type u_1} {A : Type u_2} [] [] [] [] [Algebra R A] [] [] [] (s : ) :
@[reducible]
def StarSubalgebra.commSemiringTopologicalClosure {R : Type u_1} {A : Type u_2} [] [] [] [] [Algebra R A] [] [] [] [] (s : ) (hs : ∀ (x y : { x // x s }), x * y = y * x) :
CommSemiring { x // }

If a star subalgebra of a topological star algebra is commutative, then so is its topological closure. See note [reducible non-instances].

Instances For
@[reducible]
def StarSubalgebra.commRingTopologicalClosure {R : Type u_4} {A : Type u_5} [] [] [] [Ring A] [Algebra R A] [] [] [] [] [] (s : ) (hs : ∀ (x y : { x // x s }), x * y = y * x) :
CommRing { x // }

If a star subalgebra of a topological star algebra is commutative, then so is its topological closure. See note [reducible non-instances].

Instances For
theorem StarAlgHom.ext_topologicalClosure {R : Type u_1} {A : Type u_2} {B : Type u_3} [] [] [] [] [Algebra R A] [] [] [] [] [] [Algebra R B] [] [] {S : } {φ : { x // } →⋆ₐ[R] B} {ψ : { x // } →⋆ₐ[R] B} (hφ : ) (hψ : ) (h : ) :
φ = ψ

Continuous StarAlgHoms from the topological closure of a StarSubalgebra whose compositions with the StarSubalgebra.inclusion map agree are, in fact, equal.

theorem StarAlgHomClass.ext_topologicalClosure {R : Type u_1} {A : Type u_2} {B : Type u_3} [] [] [] [] [Algebra R A] [] [] [] [] [] [Algebra R B] [] [] {F : Type u_4} {S : } [StarAlgHomClass F R { x // } B] {φ : F} {ψ : F} (hφ : ) (hψ : ) (h : ∀ (x : { x // x S }), φ (↑() x) = ψ (↑() x)) :
φ = ψ
def elementalStarAlgebra (R : Type u_1) {A : Type u_2} [] [] [] [] [] [] [Algebra R A] [] (x : A) :

The topological closure of the subalgebra generated by a single element.

Instances For
theorem elementalStarAlgebra.self_mem (R : Type u_1) {A : Type u_2} [] [] [] [] [] [] [Algebra R A] [] (x : A) :
theorem elementalStarAlgebra.star_self_mem (R : Type u_1) {A : Type u_2} [] [] [] [] [] [] [Algebra R A] [] (x : A) :

The elementalStarAlgebra generated by a normal element is commutative.

The elementalStarAlgebra generated by a normal element is commutative.

theorem elementalStarAlgebra.isClosed (R : Type u_1) {A : Type u_2} [] [] [] [] [] [] [Algebra R A] [] (x : A) :
IsClosed ↑()
theorem elementalStarAlgebra.le_of_isClosed_of_mem (R : Type u_1) {A : Type u_2} [] [] [] [] [] [] [Algebra R A] [] {S : } (hS : IsClosed S) {x : A} (hx : x S) :
theorem elementalStarAlgebra.closedEmbedding_coe (R : Type u_1) {A : Type u_2} [] [] [] [] [] [] [Algebra R A] [] (x : A) :
ClosedEmbedding Subtype.val

The coercion from an elemental algebra to the full algebra as a ClosedEmbedding.

theorem elementalStarAlgebra.starAlgHomClass_ext (R : Type u_1) {A : Type u_2} {B : Type u_3} [] [] [] [] [] [] [Algebra R A] [] [] [] [] [Algebra R B] [] {F : Type u_4} {a : A} [StarAlgHomClass F R { x // } B] {φ : F} {ψ : F} (hφ : ) (hψ : ) (h : φ { val := a, property := (_ : ) } = ψ { val := a, property := (_ : ) }) :
φ = ψ