Relative Embeddability of von Neumann Algebras and Amalgamated Free Products

In this paper we study conditions under which, for an inclusion of finite von Neumann algebras $N\subseteq M$, we have the reduced amalgamated free product ${\ast }_{N}M$ is embeddable into $(R\bar{\otimes }{N}_1{)}^{\omega }$ for some other finite von Neumann algebra $N_1$, where $R$ is the hyperfinite ${\text{II}}_1$ factor.