Separating ideals modulo the radical

Let J be a separating ideal of a unital complex Banach algebra B. Only finitely many primitive ideals of B fail to contain J, and each of them has a finite-dimensional quotient. Since the radical of B is the intersection of its primitive ideals, J embeds modulo J ∩ rad B into the finite product of these quotients. Hence J ∩ rad B has finite codimension in J, and in a semisimple Banach algebra every separating ideal is finite-dimensional. This is the finiteness half of the structure theorem for separating ideals, on which the automatic continuity of surjective homomorphisms onto, and of derivations on, semisimple Banach algebras rests.

The finiteness is proved by contradiction. An infinite family of primitive ideals not containing J contains a sequence P 0, P 1, … in which no ideal contains an earlier one: an increasing chain is ruled out because B ⧸ P 0 is finite-dimensional. Using that primitive ideals are prime and closed, and the Baire category theorem in the Banach space P n, one chooses b n ∈ P n so that each product b 0 ⋯ b (n - 1) lies outside every P m with m ≥ n. The left nest closure (b 0 ⋯ b (n - 1) J) of J then strictly decreases: its n + 1-st term lies in P n and its n-th term does not.

Main results

• Submodule.IsSeparatingIdeal.finite_setOf_isPrimitive_and_not_le: a separating ideal is contained in all but finitely many primitive ideals.
• Submodule.IsSeparatingIdeal.finiteDimensional_quotient_comap_jacobson: a separating ideal J is finite-dimensional modulo J ∩ rad B.
• Submodule.IsSeparatingIdeal.finiteDimensional_of_jacobson_eq_bot: every separating ideal of a semisimple unital complex Banach algebra is finite-dimensional.

References

• H. G. Dales, Banach Algebras and Automatic Continuity, §5.2: the structure of separating ideals.
• A. M. Sinclair, Automatic Continuity of Linear Operators, London Math. Soc. Lecture Note Series 21.

theorem Submodule.IsSeparatingIdeal.finite_setOf_isPrimitive_and_not_le {B : Type u_1} [NormedRing B] [NormedAlgebra ℂ B] [CompleteSpace B] {J : Submodule ℂ B} (hJ : J.IsSeparatingIdeal) : {P : Ideal B | P.IsPrimitive ∧ ¬J ≤ restrictScalars ℂ P}.Finite

A separating ideal lies in all but finitely many primitive ideals. For a separating ideal J of a unital complex Banach algebra B, only finitely many primitive ideals of B do not contain J.

theorem Submodule.IsSeparatingIdeal.finiteDimensional_quotient_comap_jacobson {B : Type u_1} [NormedRing B] [NormedAlgebra ℂ B] [CompleteSpace B] {J : Submodule ℂ B} (hJ : J.IsSeparatingIdeal) : FiniteDimensional ℂ (↥J ⧸ comap J.subtype (restrictScalars ℂ (Ring.jacobson B)))

A separating ideal is finite-dimensional modulo the radical. For a separating ideal J of a unital complex Banach algebra B, the quotient of J by J ∩ rad B is finite-dimensional over ℂ.

theorem Submodule.IsSeparatingIdeal.finiteDimensional_of_jacobson_eq_bot {B : Type u_1} [NormedRing B] [NormedAlgebra ℂ B] [CompleteSpace B] {J : Submodule ℂ B} (hJ : J.IsSeparatingIdeal) (hB : Ring.jacobson B = ⊥) : FiniteDimensional ℂ ↥J

Separating ideals of semisimple Banach algebras are finite-dimensional. If B is a unital complex Banach algebra with zero Jacobson radical, every separating ideal of B is finite-dimensional over ℂ.