The nilpotency of Banach nil algebras

Abstract

Theorem. If B is a Banach algebra which is also a nil algebra, then B is a nilpotent algebra.Proof. For each positive integer j, define Nj = {xQB:x' = 0}.By the theorem of Nagata-Higman [l, p. 274], it will be enough to show that some N¡ = B. Each N¡ is closed and the union of the N¡ is P. Hence, by the Baire Category Theorem, there is a fixed integer k and a fixed zQB for which Nk is a neighborhood of z. Suppose xQB. Define the B valued polynomial of a scalar variable /, p(t) = (z+t(x -z))k. Since z+t(x -z) is continuous, p(t) must equal zero for all sufficiently small /. Therefore, p(t) = 0; and, in particular p(\) = x* = 0. This completes the proof.It is easy to construct examples of normed algebras for which the above theorem is false, though the theorem is true for second category topological algebras.

Ideal-triangularizability of nil-algebras generated by positive operators

“…A is a nil-algebra, then A is ideal-triangularizable. Proof. Since A is a nil-Banach algebra, by [5] there exists n ∈ N such that A n = 0 for each A ∈ A. Ideal-triangularizability of A now follows from Theorem 2.2.…”

The Jacobson Radical for Analytic Crossed Products

“…The prime radical need not be closed; it is closed if and only if it is a nilpotent ideal (see [8] or [17,Theorem 4.4.11]). Thus for a general Banach algebra, PRad A ı PRad A ı Rad A.…”

On Nilpotency of the Separating Ideal of a Derivation

“…Let sk = lim"^oo ckn-Since ck,n = tkCk+i,n for n > k + 1 , we have sk = tksk+x . Hence sx £ CC=X Jn ■ Since fl^li Jn = {0}, it follows that sx = 0. is a closed ideal, RnS(D) is a nilpotent ideal (see [6]). Hence by Lemma 2.1 of [5], S(D) is a nilpotent ideal.…”

“…An ideal I of B is said to be nilpotent if /" -{0} for some positive integer n . It is known that every closed nil ideal is nilpotent [6].…”