Non-Hausdorff etale groupoids and C∗-algebras of left cancellative monoids

We study the question whether the representations defined by a dense subset of the unit space of a locally compact e ' tale groupoid are enough to determine the reduced norm on the groupoid C*-algebra. We present sufficient conditions for either conclusion, giving a complete answer when the isotropy groups are torsion-free. As an application, we consider the groupoid G(S) associated to a left cancellative monoid S by Spielberg and formulate a sufficient condition, which we call C*-regularity, for the canonical map Cr* (G(S)) -> Cr* (S) to be an isomorphism, in which case S has a well-defined full semigroup C*-algebra C*(S) = C*(G(S)). We give two related examples of left cancellative monoids S and T such that both are not finitely aligned and have non-Hausdorff associated e ' tale groupoids, but S is C*-regular, while T is not.