Jonathan Taylor (UP)
Renault's theorem for Cartan pairs of C*-algebras states that C*-algebras with particularly nice commutative subalgebras, called Cartan subalgebras, admit twisted groupoid models. One of the conditions that Renault required of a Cartan subalgebra was that there is a conditional expectation projecting from the ambient C*-algebra down to the subalgebra. In the groupoid setting, this corresponds to restricting functions on the groupoid to the unit space of the groupoid. In my PhD thesis, I showed that this condition can be relaxed, to allow for algebras that do not project nicely down to the subalgebra, but have conditional expectations taking values in a slightly larger algebra. Such inclusions still have twisted groupoid models, but the groupoids are no longer Hausdorff.
