  In this paper we construct faithful representations of saturated fusion
systems over discrete p-toral groups and use them to find conditions that
guarantee the existence of unitary embeddings of p-local compact groups. These
conditions hold for the Clark-Ewing and Aguad\'e-Zabrodsky p-compact groups as
well as some exotic 3-local compact groups. We also show the existence of
unitary embeddings of finite loop spaces.
