W*-rigidity paradigms for embeddings of II1 factors

I will report on a recent joint work with Sorin Popa in which we undertake a systematic study of W*-rigidity paradigms for the embedding relation between II₁ factors and their amplifications. We say that a II₁factor M stably embeds into a II₁ factor N if M may be realized as a subfactor of an amplification of N, not necessarily of finite index. This is a preorder relation and we prove that it is as complicated as it can be: under the appropriate separability assumptions, we concretely realize any partially ordered set inside the preordered class of II₁ factors with embeddability.

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