  We describe how self-adjoint ordered operator spaces, also called non-unital
operator systems in the literature, can be understood as $*$-vector spaces
equipped with a matrix gauge structure. We explain how this perspective has
several advantages over other notions of non-unital operator systems in the
literature. In particular, the category of matrix gauge $*$-vector spaces
includes injective objects and a Webster-Winkler-type duality theorem, both of
which we show generally fail with other notions of non-unital operator systems.
As applications, we characterize those subspaces of operator systems which are
kernels of completely positive maps and define a new operator space structure
on the matrix ordered dual of an operator system generalizing the classical
notion of a base norm space.
