  We study the issue of issue of purity (as a completely positive linear map)
for identity maps on operators systems and for their completely isometric
embeddings into their C$^*$-envelopes and injective envelopes. Our most general
result states that the canonical embedding of an operator system $\mathcal R$
into its injective envelope $\mathcal I(\mathcal R)$ is pure if and only if the
C$^*$-envelope of $\mathcal R$ is a prime C$^*$-algebra. To prove this, we also
show that the identity map on any AW$^*$-factor is a pure completely positive
linear map. For embeddings of operator systems into their C$^*$-envelopes, the
issue of purity is seemingly harder to describe in full generality, and so we
focus here on operator systems arising from the generators of discrete groups.
The question of purity of the identity is quite subtle for operator system that
are not C$^*$-algebras, and we have results only for two universal operator
systems arising from discrete groups. Lastly, a previously unrecorded feature
of pure completely positive linear maps is presented: every pure completely
positive linear map on an operator system $\mathcal R$ into an injective von
Neumann algebra $\mathcal M$ has a pure completely positive extension to any
operator system $\mathcal T$ that contains $\mathcal R$ as an operator
subsystem, thereby generalising a result of Arveson for the injective type I
factor $\mathcal B(\mathcal H)$.
