Embeddings of reduced free products of operator algebras

Given reduced amalgamated free products of C∗-algebras (A,φ) = ∗ ι∈I(Aι, φι) and (D,ψ) = ι∈I(Dι, ψι), an embedding A ↪ → D is shown to exist assuming there are conditional– expectation–preserving embeddings Aι ↪ → Dι. This result is extended to show the existence of the reduced amalgamated free product of certain classes of unital completely positive maps. Finally, analogues of the above mentioned results are proved for amagamated free products of von Neumann alge-bras. Introduction. The reduced free product construction, and more generally the reduced amalgamated free product construction for C∗-algebras, introduced inde-pendently by Voiculescu [21] and (somewhat less generally) Avitzour [1]