Separating ideals and primitive ideals

A separating ideal J of a unital complex Banach algebra B can act nontrivially only on finite-dimensional simple modules. Equivalently, if J is not contained in a primitive ideal P of B, then the primitive quotient B ⧸ P is finite-dimensional. This is the reduction to primitive quotients in the structure theory of separating ideals, which in turn yields the automatic continuity of surjective homomorphisms onto, and derivations on, semisimple Banach algebras.

The proof runs the right nest of J along a sequence b chosen so that the kernels on an infinite-dimensional simple Banach module E of the products b (n - 1) ⋯ b 0 strictly increase. The closure of J b (n - 1) ⋯ b 0 annihilates exactly that kernel, so the nest never stabilizes. The sequence is built from Rickart's complex Jacobson density when no element of B acts on E with nonzero finite rank; otherwise it uses elements 1 - f with f of finite rank. Algebraic simple modules are reduced to simple Banach modules by IsSimpleModule.exists_complete_norm.

Main results

• Submodule.IsSeparatingIdeal.finiteDimensional_of_isSimpleModule: a simple B-module on which some element of a separating ideal acts nontrivially is finite-dimensional.
• Submodule.IsSeparatingIdeal.finiteDimensional_quotient_of_isPrimitive: if a separating ideal is not contained in a primitive ideal P, then B ⧸ P is finite-dimensional.

References

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

theorem Submodule.IsSeparatingIdeal.finiteDimensional_of_isSimpleModule {B : Type u_1} [NormedRing B] [NormedAlgebra ℂ B] [CompleteSpace B] {M : Type u_2} [AddCommGroup M] [Module B M] [Module ℂ M] [IsScalarTower ℂ B M] [IsSimpleModule B M] {J : Submodule ℂ B} (hJ : J.IsSeparatingIdeal) {j : B} (hj : j ∈ J) {m : M} (hjm : j • m ≠ 0) : FiniteDimensional ℂ M

Separating ideals act only on finite-dimensional simple modules. Let J be a separating ideal of a unital complex Banach algebra B and M a simple left B-module, with no topology assumed. If some element of J acts nontrivially on M, then M is finite-dimensional over ℂ.

theorem Submodule.IsSeparatingIdeal.finiteDimensional_quotient_of_isPrimitive {B : Type u_1} [NormedRing B] [NormedAlgebra ℂ B] [CompleteSpace B] {J : Submodule ℂ B} (hJ : J.IsSeparatingIdeal) {P : Ideal B} (hP : P.IsPrimitive) (hJP : ¬J ≤ restrictScalars ℂ P) : FiniteDimensional ℂ (B ⧸ P)

Separating ideals have finite-dimensional primitive quotients. If a separating ideal J of a unital complex Banach algebra B is not contained in a primitive ideal P, then B ⧸ P is finite-dimensional over ℂ.