Boundary quotients of C${}^{\ast }$-algebras of left cancellative monoids and their groupoid models

For a left cancellative monoid $S$ we consider a quotient of the reduced semigroup C${}^{\ast }$-algebra $C_r^{\ast }(S)$ known as the boundary quotient. We present two potential groupoid models for this boundary quotient, obtained as reductions of Paterson and Spielberg’s groupoids associated to $S$, and formulate conditions on $S$ which guarantees that either is a groupoid model. We outline how these conditions are related to the notions (strong) C${}^{\ast }$-regularity introduced in a previous paper, and construct an example of a left cancellative monoid which is not C${}^{\ast }$-regular, but satisfies both of the new conditions.