Prime-like Elements and Semi-direct Products in Commutative Banach Algebras

Abstract

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 sequencefor 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 BACKGROUND AND NOTATIONThis 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 44

The multiplier algebra and BSE-functions for certain product of Banach algebras

“…Indeed, in this case the product of two elements (b, a) and (b ′ , a ′ ) of B ⊕ I is given by (b, a)(b ′ , a ′ ) = (bb ′ , aa ′ + ba ′ + ab ′ ), and this algebra endowed with the norm (b, a) = b + a is a Banach algebra. This notion was also studied by Berndt [4] and Thomas [21,22] and they considered under what conditions a commutative Banach algebra is the semidirect product of a subalgebra and a principal ideal. We also note that the algebra B ⊕ I is a splitting extension of B by I.…”

An Almost Prime Element in a Commutative Radical Banach Algebra

“…The above definition can be extended to define (weakly) almost k-prime and (weakly) almost prime elements (see below). These definitions are due to Thomas; see [3]. Thomas, with help from a theorem of Runde, has reduced the unbounded Kleinecke-Shirokov conjecture [1] to showing the non-existence of elements with certain prime-like properties in commutative radical Banach algebras [4].…”