instContinuousLinearMapClassComplex

English

Any nonunital star-algebra homomorphism between complex C*-algebras naturally carries a structure of a continuous linear map over the complex numbers.

Русский

Любой неконглуный звездно-алгебраический гомоморфизм между комплексными C*-алгебрами естественно наделяет структуру непрерывного линейного отображения над комплексными числами.

LaTeX

$$$\text{φ is a continuous linear map over } \mathbb{C}$$$

Lean4

/-- Non-unital star algebra homomorphisms between C⋆-algebras are continuous linear maps.
See note [lower instance priority] -/
theorem instContinuousLinearMapClassComplex : ContinuousLinearMapClass F ℂ A B :=
  { NonUnitalAlgHomClass.instLinearMapClass with
    map_continuous := fun φ =>
      AddMonoidHomClass.continuous_of_bound φ 1 (by simpa only [one_mul] using nnnorm_apply_le φ) }

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