I will begin with a brief discussion of Renault's classical construction of a groupoid that gives rise to the Cuntz algebra O_n with its canonical Cartan subalgebra. Renault's groupoid model has isotropy (i.e. the dynamics has some fixed points). This means that the canonical Cartan in O_n isn't a C*-diagonal, and in particular has infinite diagonal dimension.
In the second half of the talk, I will outline my recent work with Philipp Sibbel on constructing a groupoid model for the Cuntz algebra O_n with no isotropy (i.e. a principal groupoid). This groupoid gives rise to a C*-diagonal inside of O_n whose spectrum is a Cantor space.
The groupoid can be realised as a product of the Deaconu-Renault groupoid associated to a well-chosen topological graph with the groupoid of a well-chosen AF equivalence relation.
