We develop results which show that elements in the radical of a commutative Banach algebra are often precluded from having prime-like properties if we avoid certain exceptional situations involving torsion elements. This makes the proof of the Singer-Wermer conjecture conceptually much clearer. It also motivates the definition of an element having regular powers and allows us to strengthen our previous results concerning necessary conditions for a commutative Banach algebra A to be the semidirect product of some subalgebra together with a specified principal ideal sA > , or, equivalently, concerning necessary conditions for there to be an algebraic splitting of the short exact sequence
for some given element s in A. In particular, we show that if A is a radical Banach algebra and s has regular powers then no such splitting is possible.
1997 Academic Press
1. BACKGROUND AND NOTATION
This paper continues an investigation of the structure of the radical of a commutative Banach algebra which was pioneered by G. R. Allan [1] and J. Esterle [6], and which the author continued in [11] and [12]. Our attempt here is both to unify some of the existing results by focusing our attention on non-nilpotent elements with prime-like properties in the radical, and to improve existing theorems, such as those in [12]. Our main theorem is Theorem 4.1 in Section 4; this section also contains some interesting examples.
We need to make some definitions and explain our notation. Let A denote a commutative algebra over the complex field. For the following definitions no topology is needed, but we will eventually specialize to the article no. FU963020