Embeddings of reduced free products of operator algebras

TL;DR

This paper establishes the existence of embeddings for reduced free products of operator algebras and extends these results to certain classes of maps and von Neumann algebras, advancing understanding of their structural relationships.

Contribution

It introduces new embedding results for reduced free products of C*-algebras and extends these to unital completely positive maps and von Neumann algebras.

Findings

Embeddings of reduced free products exist under certain conditions.

Extensions to unital completely positive maps are demonstrated.

Analogues for von Neumann algebras are proved.

Abstract

Given reduced amalgamated free products of C$^*$-algebras, $(A,phi)=*_i(A_i,phi_i)$ and $(D,psi)=*_i(D_i,psi_i)$, an embedding $A\to D$ is shown to exist assuming there are conditional expectation preserving embeddings $A_i\to D_i$. This result is extended to show the existance of the reduced amalgamated free product of certain classes of unital completely positive maps. Analogues of the above mentioned results are proved for von Neumann algebras.