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Baire Category And Laurent Extensions

Published online by Cambridge University Press:  20 November 2018

Daniel R. Farkas
Daniel R. Farkas*
Affiliation:
Virginia Polytechnic Institute and State University, Blacksburg, Virginia
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Based on a strategy of Kaplansky ([3]), Dixmier proved that a prime, separable C*-algebra is primitive ([1]). As a consequence, when the C*-closure of a countable discrete group is prime, it is primitive. The argument may be regarded as a clever application of the Baire Category Theorem to the spectrum of irreducible representations.

The present note is the first step in adapting this technique to abstract group algebras. For which groups G is the primitive ideal space of k[G] a Baire space? One corollary of our main result is that the space is Baire when k is an uncountable field and G is a polycyclic-by-finite group. This gives an alternate proof of a special case of Passman's theorem that such a k[G] will be primitive when its center is k ([4], p. 379).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 4 , 01 August 1979 , pp. 824 - 830
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Dixmier, J., Sur les C*-algèbres, Bull. Soc. Math. Franc. 88 (1960), 95–112.Google Scholar
2. Goldie, A. and Michler, G., Ore extensions and polycyclic group rings, J. London Math. Soc. (2) 9 (1974/75), 337–345.Google Scholar
3. Kaplansky, I., The structure of certain operator algebras, T.A.M.S. 70 (1951), 219–255.Google Scholar
4. Passman, D. S., The algebraic structure of group rings (Wiley-Interscience, N.Y., 1977).Google Scholar