# Simplicity of reduced C*-algebras for non-Hausdorff etale groupoids

It is known that for a Hausdorff locally compact etale groupoid, the reduced C*-algebra is simple iff the groupoid is minimal (meaning the orbit of each unit is dense) and topologically principal (meaning the set of units with trivial isotropy is dense).

If one considers the case where only the unit space is Hausdorff, then it is known the above two conditions are not sufficient.

Question: Are there simple to state necessary and sufficient conditions for simplicity of the reduced C*-algebra in the non-Hausdorff setting?

I suspect something can be extracted from the Khoshkam-Skandalis paper in Crelle's journal on regular representations of non-Hausdorff groupoids but I am not an operator theorist by trade.

Edit: I omitted the hypothesis that the groupoid be amenable. This should be added to both the background and the question.

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## 1 Answer

Simplicity of the reduced groupoid C*-algebra of a locally compact Hausdorff etale groupoid does not imply that the groupoid is topological principal. The standard counter-example is the free group on 2 generators - the reduced group C*-algebra is known to be simple, a result obtained first by Powers. As far as I know, the best 'iff' statement is that when there is at least a single unit with trivial isotropy group, the reduced C*-algebra is simple iff the groupoid is minimal.

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Sorry, I forgot to add the hypothesis that the groupoid be amenable. –  Benjamin Steinberg Oct 11 '12 at 1:32