Idempotents in the separating space of a homomorphism

Let θ : A → B be a homomorphism of complex Banach algebras, not assumed continuous. An algebra homomorphism can only shrink quasispectra, so the spectral radius of θ x is at most ‖x‖. This rules out nonzero idempotents of B that lie both in the range of θ and in its separating space: if θ (a n) → p = θ u with a n → 0, then the elements θ (u * a n * u) lie in the corner p B p, tend to p, and have spectral radii tending to zero, which is impossible near a nonzero idempotent. In particular, the separating space of a surjective homomorphism contains no nonzero idempotent.

The structure theory of separating ideals produces a nonzero idempotent in every nonzero finite-dimensional separating ideal of a semisimple algebra; this obstruction is what excludes it for surjective homomorphisms.

Main results

• NonUnitalAlgHom.eq_zero_of_isIdempotentElem_of_mem_separatingSpace: an idempotent in the range and in the separating space of θ is zero.

References

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

theorem NonUnitalAlgHom.eq_zero_of_isIdempotentElem_of_mem_separatingSpace {A : Type u_1} {B : Type u_2} [NonUnitalNormedRing A] [NormedSpace ℂ A] [IsScalarTower ℂ A A] [SMulCommClass ℂ A A] [CompleteSpace A] [NonUnitalNormedRing B] [NormedSpace ℂ B] [IsScalarTower ℂ B B] [SMulCommClass ℂ B B] [CompleteSpace B] (θ : A →ₙₐ[ℂ] B) {p : B} (hp : IsIdempotentElem p) (hpθ : p ∈ Set.range ⇑θ) (hpS : p ∈ (↑θ).separatingSpace) : p = 0

Separating spaces of homomorphisms contain no idempotents from the range. Let θ : A → B be a homomorphism of complex Banach algebras, not assumed continuous. An idempotent p of B lying both in the range of θ and in the separating space of θ is zero. For surjective θ, the separating space contains no nonzero idempotent.