Automatic continuity of surjective homomorphisms onto semisimple Banach algebras

Johnson's theorem: a surjective homomorphism θ : A → B of complex Banach algebras, not assumed continuous, is continuous whenever B is unital and semisimple, meaning that its Jacobson radical is zero. The algebra A need not be unital.

The separating space of θ is a separating ideal of B, hence finite-dimensional because B is semisimple. A nonzero finite-dimensional ideal of a semisimple algebra contains a nonzero idempotent, and the separating space of θ contains no nonzero idempotent from the range of θ. Since θ is surjective, the separating space is zero, and θ is continuous.

For a nonunital B with zero Jacobson radical, θ extends to a surjective homomorphism of the ℓ¹ unitizations, whose target is again semisimple, and continuity of the extension restricts to θ. Surjectivity cannot be dropped when B is noncommutative.

Main results

• NonUnitalAlgHom.continuous_of_surjective_of_jacobson_eq_bot: Johnson's theorem.
• NonUnitalAlgHom.continuous_of_surjective_of_nonUnitalJacobson_eq_bot: Johnson's theorem for a nonunital semisimple target.
• NonUnitalAlgHom.isHomeomorph_of_bijective_of_nonUnitalJacobson_eq_bot: a bijective homomorphism onto a semisimple, possibly nonunital, Banach algebra is a homeomorphism.
• AlgEquiv.isHomeomorph_of_jacobson_eq_bot: an algebra isomorphism from a complex Banach algebra onto a semisimple unital complex Banach algebra is a homeomorphism.

References

• B. E. Johnson, The uniqueness of the (complete) norm topology, Bull. Amer. Math. Soc. 73 (1967), 537–539.
• H. G. Dales, Banach Algebras and Automatic Continuity, §5.2, in particular the structure of separating ideals in semisimple Banach algebras and Johnson's uniqueness-of-norm theorem.

theorem NonUnitalAlgHom.continuous_of_surjective_of_jacobson_eq_bot {A : Type u_1} {B : Type u_2} [NonUnitalNormedRing A] [NormedSpace ℂ A] [IsScalarTower ℂ A A] [SMulCommClass ℂ A A] [CompleteSpace A] [NormedRing B] [NormedAlgebra ℂ B] [CompleteSpace B] (θ : A →ₙₐ[ℂ] B) (hθ : Function.Surjective ⇑θ) (hB : Ring.jacobson B = ⊥) : Continuous ⇑θ

Johnson's theorem. A surjective homomorphism θ : A → B of complex Banach algebras, not assumed continuous, is continuous when B is unital with zero Jacobson radical.

theorem NonUnitalAlgHom.continuous_of_surjective_of_nonUnitalJacobson_eq_bot {A : Type u_1} [NonUnitalNormedRing A] [NormedSpace ℂ A] [IsScalarTower ℂ A A] [SMulCommClass ℂ A A] [CompleteSpace A] {B : Type u_3} [NonUnitalNormedRing B] [NormedSpace ℂ B] [IsScalarTower ℂ B B] [SMulCommClass ℂ B B] [CompleteSpace B] (θ : A →ₙₐ[ℂ] B) (hθ : Function.Surjective ⇑θ) (hB : TauCeti.nonUnitalJacobson ℂ B = ⊥) : Continuous ⇑θ

Johnson's theorem, nonunital form. A surjective homomorphism θ : A → B of complex Banach algebras, not assumed continuous, is continuous when B has zero Jacobson radical. Neither algebra need be unital.

theorem NonUnitalAlgHom.isHomeomorph_of_bijective_of_nonUnitalJacobson_eq_bot {A : Type u_1} [NonUnitalNormedRing A] [NormedSpace ℂ A] [IsScalarTower ℂ A A] [SMulCommClass ℂ A A] [CompleteSpace A] {B : Type u_3} [NonUnitalNormedRing B] [NormedSpace ℂ B] [IsScalarTower ℂ B B] [SMulCommClass ℂ B B] [CompleteSpace B] (θ : A →ₙₐ[ℂ] B) (hθ : Function.Bijective ⇑θ) (hB : TauCeti.nonUnitalJacobson ℂ B = ⊥) : IsHomeomorph ⇑θ

A bijective homomorphism θ : A → B of complex Banach algebras, not assumed continuous, is a homeomorphism when B has zero Jacobson radical. Neither algebra need be unital. The inverse is continuous by the open mapping theorem.

theorem AlgEquiv.isHomeomorph_of_jacobson_eq_bot {A : Type u_1} {B : Type u_2} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [NormedRing B] [NormedAlgebra ℂ B] [CompleteSpace B] (e : A ≃ₐ[ℂ] B) (hB : Ring.jacobson B = ⊥) : IsHomeomorph ⇑e

An algebra isomorphism from a complex Banach algebra onto a unital complex Banach algebra with zero Jacobson radical, not assumed continuous, is a homeomorphism. The inverse is continuous by the open mapping theorem.