I will report on a recent joint work with Sorin Popa in which we undertake a systematic study of W*-rigidity paradigms for the embeddability relation between II_1 factors and their amplifications. We say that a II_1 factor M stably embeds into a II_1 factor N if M may be realized as a subfactor of an amplification of N, not necessarily of finite index. We provide families of II_1 factors that are mutually non stably embeddable, as well as families that are mutually embeddable, yet nonisomorphic. We prove that the preorder relation of stable embeddability is as complicated as it can be since it contains any partially ordered set. We also obtain numerous computations of invariants of II_1 factors, including descriptions of all stable self embeddings, outer automorphism groups, etc.

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