All Primes Have Closed Range.

In this paper, we show first that any prime (or semiprime) element of a commutative Banach algebra must have closed range. As a corollary, we find that in a commutative radical Banach algebra, all primes are zero divisors; indeed, all semiprimes are zero divisors (see below for the definition of semiprimeness). Our result is also true of a semiprime that is in the centre of a noncommutative Banach algebra.The proof is fairly simple and entertaining, and we obtain a result that is helpful for the ambitious classification of elements in commutative radical Banach algebras being attempted by Marc Thomas. It is also related to the unbounded Kleinecke–Shirov conjecture. [ABSTRACT FROM PUBLISHER]