An Introduction to Non-Hausdorff Groupoids

For 2nd-countable effective locally compact Hausdorff etale groupoids, it is known that every nonzero ideal of the reduced C*-algebra has nonzero intersection with the continuous functions on the unit space that vanish at infinity. In this talk, I will discuss an example of Exel using the groupoid of germs construction where this property fails. Showing this involves constructing an example of a special kind of function on the groupoid whose "open" support has empty interior. For any 2nd-countable effective locally compact etale groupoid, a paper of Clark, Exel, Pardo, Sims, and Starling shows that the existence of such an element guarantees that the reduced C*-algebra is not simple.