Course Description
The hyperfinite \(II_1\) factor \(R\) has played a central role in operator algebras ever since Murray and von Neumann introduced it, some 75 years ago. It is the unique amenable \(II_1\) factor (Connes 1976), and in some sense the smallest, as it can be embedded in multiple ways in any other \(II_1\) factor \(M\). Many problems in operator algebras could be solved by constructing ”ergodic” such embeddings \(R \to M\). I will revisit such results and applications, through a new perspective, which emphasizes the decomposition \(M\) as a Hilbert bimodule over \(R\). I will prove that any \(II_1\) factor Madmits coarse embeddings of \(R\), where the orthocomplement of \(R\) in \(M\) is a multiple of \(L^2(R) \overline{\otimes} L^2(R^{op})\). I will also prove that in certain situations, \(M\) admits tight embeddings of \(R\). Finally, I will revisit some well known open problems, and propose some new ones, through this perspective.