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Minimal primal ideals in Banach algebras

Published online by Cambridge University Press:  24 October 2008

Douglas W. B. Somerset
Douglas W. B. Somerset
Affiliation:
Thackit Eaves, Highclere, Newbury, Berkshire, RG15 9QU
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Abstract

An ideal I in a ring R is primal if whenever J1, …, Jn is a finite set of ideals of R with J1 … Jn = {0} then Ji ⊆ I for at least one i ∈ {1, …, n}. If the ring is commutative then is is easily shown that each primal ideal contains a prime ideal. In this paper it is shown that in a separable semi-simple Banach algebra each primal ideal contains a prime ideal, and that the space of minimal prime ideals is compact and extremally disconnected in the hull-kernel topology. An example is given of an inseparable C-algebra with a primal ideal which does not contain a prime ideal.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 115 , Issue 1 , January 1994 , pp. 39 - 52
Copyright
Copyright © Cambridge Philosophical Society 1994

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