The hyperfinite II₁ factor R has played a central role in operator algebras ever since Murray and von Neumann introduced it in 1936-1943. It is the smallest II₁ factor, as it can be embedded in multiple ways in any other II1₁ factor M, and the unique amenable II₁ factor (Connes 1976). I have shown in 1981 that R can be embedded ergodically into any separable II₁ factor. I will discuss two new results I have obtained, along these lines :

1. Any separable II₁ factor M admits coarse embeddings of R, i.e., an embedding R↪ M such that L²M⊖L²R is a multiple of the coarse Hilbert R-bimodule L²R⊗L²Rᵒᵖ (equivalently, left-right multiplication by R on L²M⊖L²R gives a normal representation of R⊗Rᵒᵖ ).

2. Any separable II₁ factor admits an ergodic embedding of R.