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Bulletin of the London Mathematical Society Advance Access originally published online on February 20, 2009
Bulletin of the London Mathematical Society 2009 41(2):354-366; doi:10.1112/blms/bdp009
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© 2009 London Mathematical Society

Uncountable families of prime z-ideals in C0(R)

Hung Le Pham

Department of Mathematical and Statistical Sciences
University of Alberta
Edmonton
Canada AB T6G 2G1

Received 22 January 2008. Revision received 29 October 2008.

Denote by c = 2N0 the cardinal of continuum. We construct an intriguing family (P{alpha}: {alpha} isin c) of prime z-ideals in C0 (R) with the following properties:

  1. if f isin Pi0 for some i0 isin c, then F isin Pi for all but finitely many i isin c;
  2. {cap} i!=i0 Pi {nsubset} Pi0 for each i0 isin c.
As an application, assuming the continuum hypothesis, there exists a homomorphism from C0 (R) into a Banach algebra whose continuity ideal is not the intersection of any countable family of prime ideals in C0 (R).

We also construct a well-ordered increasing chain, as well as a well-ordered decreasing chain, of order type {kappa} of prime z-ideals in C0 (R) for any ordinal {kappa} of cardinality c.

2000 Mathematics Subject Classification 46J10 (primary), 54C40, 46J20 (secondary).

This research is supported by a Killam Postdoctoral Fellowship and an honorary PIMS PDF.


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