# 30 - The ubiquitous hyperfinite II1 factor, 2019

Top Global Course Special Lectures 1 ## The ubiquitous hyperfinite II1 factor
April 8, 9, 10, 11 and 12, 2019 |

#### Course Description

The hyperfinite factor 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 factor (Connes 1976), and in some sense the smallest, as it can be embedded in multiple ways in any other factor . Many problems in operator algebras could be solved by constructing ''ergodic'' such embeddings ↪. I will revisit such results and applications, through a new perspective, which emphasizes the decomposition as a Hilbert bimodule over . I will prove that any factor admits coarse embeddings of , where the orthocomplement of in is a multiple of . I will also prove that in certain situations, admits tight embeddings of . Finally, I will revisit some well known open problems, and propose some new ones, through this perspective.

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