Baire Category And Laurent Extensions

Abstract

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).