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ALL PRIMES HAVE CLOSED RANGE

Published online by Cambridge University Press:  14 June 2001

C. J. READ
C. J. READ
Affiliation:
Faculty of Mathematics, University of Leeds, Leeds LS2 9JT, read@amsta.leeds.ac.uk
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Abstract

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.

Type
NOTES AND PAPERS
Information
Bulletin of the London Mathematical Society , Volume 33 , Issue 3 , May 2001 , pp. 341 - 346
Copyright
© The London Mathematical Society 2001

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