English
The preimage (comap) of a nonunital star-subalgebra under a nonunital star algebra homomorphism is again a nonunital star-subalgebra.
Русский
Предобраз неединичной звёздной подпалгебры под непринадующим гомоморфизмом сохраняет структуру неединичной звёздной подпалгебры.
LaTeX
$$comap(f,S) is a nonunital star-subalgebra of A; equivalently, comap f S = { a ∈ A | f(a) ∈ S }.$$
Lean4
/-- Preimage of a non-unital star subalgebra under a non-unital star algebra homomorphism. -/
def comap (f : F) (S : NonUnitalStarSubalgebra R B) : NonUnitalStarSubalgebra R A
where
toNonUnitalSubalgebra := S.toNonUnitalSubalgebra.comap f
star_mem' := @fun a (ha : f a ∈ S) => show f (star a) ∈ S from (map_star f a).symm ▸ star_mem (s := S) ha
