  Consider an Archimedean partially ordered vector space $X$ with generating
cone (or, more generally, a pre-Riesz space $X$). Let $P$ be a linear
projection on $X$ such that both $P$ and its complementary projection $I - P$
are positive; we prove that the range of $P$ is a band. This shows that the
well-known concept of band projections on vector lattices can, to a certain
extent, be transferred to the framework of ordered vector spaces.
