The Primitive Ideal Space of Two-Step Nilpotent Group C*-Algebras

Let N be a two-step nilpotent, locally compact, second countable group having center Z and quotient A = N/Z. We study the Jacobson topology on the primitive ideal space Prim C*(N) of the group C*-algebra of N. We are able to describe this topology in terms of convergence of subgroup-representation pairs, as used by the first author in an earlier work. Under appropriate conditions on N, we are able to describe Prim C*(N) globally as the quotient of a principal  bundle over Ẑ modulo an equivalence relation determined entirely by the group structure. We use this second result to compute the primitive ideal spaces of several examples, including all finitely generated, non-torsion two-step nilpotent discrete groups of rank less than or equal to five. Applications of our methods to more general central twisted crossed products are discussed.