  We initiate a study of non-commutative Choquet boundary for spaces of
unbounded operators. We define the notion of local boundary representations for
local operator systems in locally C$^*$-algebras and prove that local boundary
representations provide an intrinsic invariant for a particular class of local
operator systems. An appropriate analog of purity of local completely positive
maps on local operator systems is used to characterize local boundary
representations for local operator systems in Frechet locally C$^*$-algebras.
