On Ideals That Are Closed Under Continuous Endomorphisms

Abstract

Let 𝔽 be the field ℝ or ℂ, and let 𝔽 ⟨ X ⟩ be the free associative algebra generated by the infinite set X. If f ∈ 𝔽 ⟨ X ⟩, define its norm ‖f‖ as the sum of the absolute value of its coefficients. We describe the ideals of 𝔽 ⟨ X ⟩ which are closed under all continuous endomorphisms of 𝔽 ⟨ X ⟩. An element f in the completion is called a series identity for a normed algebra A if f belongs to the intersection of the kernels of all continuous homomorphisms from to A. We describe such identities for a nil Banach algebra.