Groups with T1 primitive ideal spaces

AbstractWe give a simple necessary and sufficient condition for the group C∗-algebra of a connected locally compact group to have a T1 primitive ideal space, i.e., to have the property that all primitive ideals are maximal. A companion result settles the same question almost entirely for almost connected groups. As a by-product of the method used, we show that every point in the primitive ideal space of the group C∗-algebra of a connected locally compact group is at least locally closed. Finally, we obtain an analog of these results for discrete finitely generated groups; in particular the primitive ideal space of the group C∗-algebra of a discrete finitely generated solvable group is T1 if and only if the group is a finite extension of a nilpotent group