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

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 de nition 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 speci ed principal ideal sA, or, equivalently, concerning necessary conditions for there to be an algebraic splitting of the short exact sequence 0! sA !A! (A=sA)! 0 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. x1. 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 de nitions and explain our notation. Let A denote a commutative algebra over the complex eld. For the following de nitions no topology is needed, but we will eventually specialize to the case where A is a commutative Banach algebra. We will use the terms subspace, subalgebra, ideal, and homomorphism in the algebraic sense, i.e. even if there is a topology on the algebra, we y The author would like to thank NSERC and the department of Mathematics at the University of Manitoba for support in writing this paper.