W*-rigidity paradigms for embeddings of II${}_1$ factors

Abstract

We undertake a systematic study of W*-rigidity paradigms for the embeddability relation $\hookrightarrow$ between separable II${}_1$ factors and its stable version ${\hookrightarrow }_s$, obtaining large families of non stably isomorphic II${}_1$ factors that are mutually embeddable and families of II${}_1$ factors that are mutually non stably embeddable. We provide an augmentation functor $G\mapsto H_G$ from the category of groups into icc groups, so that $L(H_{G_1}){\hookrightarrow }_{s}L(H_{G_2})$ iff $G_1\hookrightarrow G_2$. We construct complete intervals of II${}_1$ factors, including a strict chain of II${}_1$ factors $(M_k{)}_{k\in \mathbb{Z}}$ with the property that if $N$ is any II${}_1$ factor with $M_i{\hookrightarrow }_{s}N$ and $N{\hookrightarrow }_s{M}_j$, then $N\cong M_k^t$ for some $i\le k\le j$ and $t>0$.

v3: minor changes, final version, to appear in Communications in Mathematical Physics

References in corpus (1)

• There is no separable universal II_1-factor

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