Irreducible representation of $C^*(D_\infty)$, group $C^*$-algebra of an infinite dihedral group

Question

I have a question about an irreducible representation of the (full) group $C^*$-algebra of an infinite dihedral group $D_\infty$, denoted by $C^*(D_\infty)$.

Ultimately, I'm interested in finding a primitive ideal space of $C^*(D_\infty)$ which is the kernel of irreducible representation of $C^*(D_\infty)$.

But I'm having a hard time finding it.

$C^*(D_\infty)$ is isomorphic to $A$, the universal $C^*$-algebra generated by two projections, $p,q$.

So say $\pi$ is a representation of $A$ on some Hilbert space, $H$.

Then $\pi(p)$ and $\pi(q)$ are projections on $H$.

So using this fact, should I consider an irreducible representation on $A$ first?

Or should I find pure states on $C^*(D_\infty)$ first since they correspond to irreducible representations of $C^*(D_\infty)$?

I'm fairly new to this stuff, so any reference will be definitely appreciated.

And thank you in advance.

Answer 1

All answers can be found in that paper by Raeburn and Sinclair: www.mscand.dk/article/download/12283/10299

The authors do not claim originality, as they collect and smooth out results known for a certain amount of time; indeed I remember doing myself the exercise of writing down the structure of $C^*(D_\infty)$ around 1982, based on the papers of Dixmier (1948) and Halmos (1969).