In this paper we study conditions under which, for an inclusion of finite von Neumann algebras N ⊆ M, we have the reduced amalgamated free product N M is embeddable into (R ⊗̄ N₁)^ω for some other finite von Neumann algebra N₁, where R is the hyperfinite II₁ factor.
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Gao et al. (2020) studied this question.