  We generalize a result of T. Koberda by showing that the natural action of
the automorphism group on the space of left-orderings is faithful for all
nonabelian bi-orderable groups G, as well as for a certain class of
left-orderable groups that includes the braid groups. As a corollary we show
that the action of the automorphism group of G on the boundary of G is faithful
whenever G is bi-orderable and hyperbolic. We also analyze the action of the
commensurator of G on its space of virtual left-orderings.
