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Is there a general way of finding a primitive ideal space of $C^*$-algebra?

For example, if $C^*$-algebra is given by the universal $C^*$-algebra generated by two self-adjoint unitary elements, how does one determine its primitive ideal space?

What is a general (or best) apprach to finding its primitive ideal space of a $C^*$-algebra?

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    $\begingroup$ The example you consider is very special, as it is the $C^*$-algebra of the infinite dihedral group, which is notoriously type I. The spectrum in that case is "an interval with 2+2 endpoints", i.e. the union of two closed intervals identified along their interiors. This well-known fact can be found e.g. in these notes by I. Putnam: math.uvic.ca/faculty/putnam/t/Math_533_Lecture_Notes.pdf $\endgroup$
    – Alain Valette
    Commented Apr 21, 2015 at 21:42
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    $\begingroup$ Your general question seems too optimistic and open-ended, although as @AlainValette has pointed out one can do quite a lot for Type I groups. You will have problems if your Cstar algebra is not "postliminal", a term which is explained and studied in e.g. Dixmier's book. $\endgroup$
    – Yemon Choi
    Commented Apr 22, 2015 at 11:48
  • $\begingroup$ sometimes the following fact is useful: If $I$ is a closed ideal in a $C^*$-algebra $A$, it is $$\hat{A}=\hat{A}_I\coprod \hat{A}_{A/I}=\widehat{I}\coprod \widehat{A/I},$$ where $\hat{A}_I=\{[\pi]\in \hat{A}: \pi(I)\neq 0\}$ and $\hat{A}_{A/I}=\{[\pi]\in \hat{A}: \pi(I)=0\}$. I used it in this mathoverflow.net/questions/219419/… case. But as the others said, your question is too open ended. $\endgroup$
    – Sabrina Gemsa
    Commented May 24, 2016 at 6:07
  • $\begingroup$ Just to update the link in Alain's comment: math.uvic.ca/faculty/putnam/ln/lecture_notes.html $\endgroup$
    – Yemon Choi
    Commented Dec 12, 2016 at 18:59

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