The class of Type I C*-algebras was prominently studied in the early days of C*-algebra theory due to their particularly nice representation theory, but for more than half a century, it has been clear that this case is a rather special one, and consequently the subject has been focused squarely within the non-Type I case.
Thus it may come as a surprise that the study of structure-preserving isomorphisms of unital Type I graph C*-algebras leads to deep questions and useful applications. These are exactly the C*-algebras associated to graphs such as having the properties that they have finitely many vertices, and support only a countable number of infinite paths.
I will give an overview of what is meant by and known about “structure-preserving isomorphisms” here, and detail a few examples of particular importance arising from the theory of quantum spaces, all the while I try to convince the audience that in this case, the theory for graph C*-algebras and for Leavitt path algebras should be completely parallel. All new results come from ongoing joint projects with Ruiz and with Mikkelsen.