English
The composition of two nonunital ⋆-algebra homomorphisms is itself a nonunital ⋆-algebra homomorphism.
Русский
Композиция двух ненулевых ⋆-алгоморфизмов образует вновь ненуль-⋆-алгоморфизм.
LaTeX
$$$f : B \to\star_n_a[R] C$ and $g : A \to\star_n_a[R] B$ imply $f \circ g : A \to\star_n_a[R] C$ with $(f \circ g)(a) = f(g(a))$$$
Lean4
/-- The composition of non-unital ⋆-algebra homomorphisms, as a non-unital ⋆-algebra
homomorphism. -/
def comp (f : B →⋆ₙₐ[R] C) (g : A →⋆ₙₐ[R] B) : A →⋆ₙₐ[R] C :=
{ f.toNonUnitalAlgHom.comp g.toNonUnitalAlgHom with
map_star' := by
simp only [map_star, NonUnitalAlgHom.toFun_eq_coe, NonUnitalAlgHom.coe_comp, coe_toNonUnitalAlgHom,
Function.comp_apply, forall_const] }
