The ubiquitous hyperfinite ${\rm II}_1$ factor

2019/04/08 Mon 15:00 - 17:00
2019/04/09 Tue 15:00 - 17:00
2019/04/10 Wed 13:00 - 15:00
2019/04/11 Thu 15:00 - 17:00
2019/04/12 Fri 15:00 - 17:00
Sorin Popa
Kyoto University / UCLA

The hyperfinite ${\rm II}_1$ 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 ${\rm II}_1$ factor (Connes 1976), and in some sense the smallest, as it can be embedded in multiple ways in any other ${\rm II}_1$ factor $M$. Many problems in operator algebras could be solved by constructing ''ergodic'' such embeddings $R \hookrightarrow M$. 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 ${\rm II}_1$ factor $M$ admits coarse embeddings of $R$, where the orthocomplement of $R$ in $M$ is a multiple of $L^2(R) \,\overline{\otimes}\, L^2(R^{\rm op})$. 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.