## The ubiquitous hyperfinite II1 factor

### Lecture 1Sorin POPA

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.

### Details

Year/Term
2019 / Intensive, First semester
Date
April 8th to April 12th, 2019
Faculty/