Purity of the embeddings of operator systems into their C∗- and injective envelopes

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