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 'etale groupoid are enough to determine the reduced norm on the groupoid C${}^{\ast }$-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${}^{\ast }$-regularity, for the canonical map $C_r^{\ast }(G(S))\to C_r^{\ast }(S)$ to be an isomorphism, in which case $S$ has a well-defined full semigroup C${}^{\ast }$-algebra $C^{\ast }(S)=C^{\ast }(G(S))$. We give two related examples of left cancellative monoids $S$ and $T$ such that both are not finitely aligned and have non-Hausdorff associated 'etale groupoids, but $S$ is C${}^{\ast }$-regular, while $T$ is not.