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

We undertake a systematic study of W-rigidity paradigms for the embeddability relation →֒ between separable II1 factors and its stable, weaker version →֒s involving amplifications of factors. We obtain concrete large families of non stably isomorphic II1 factors that are mutually embeddable (many-to-one paradigm) and large families of II1 factors that are mutually non stably embeddable (disjointness paradigm). We provide an augmentation functor G 7→ HG from the category of groups into icc groups, so that L(HG1) →֒s L(HG2) iff G1 →֒ G2. We produce a large class of II1 factors for which we compute all stable self-embeddings, including II1 factors M without nontrivial endomorphisms M →֒ M t and II1 factors with numerous prescribed outer automorphism groups. We construct concrete complete intervals of II1 factors, indexed over a large variety of partially ordered sets, including a strict chain of II1 factors (Mk)k∈Z with the property that if N is any II1 factor with Mi →֒s N and N →֒s Mj, then N ∼=M t k for some i ≤ k ≤ j and t > 0.

[1]  Intervals in subgroup lattices of infinite groups , 1989 .

[2]  S. Popa,et al.  On a class of $\mathrm{II}_1$ factors with at most one Cartan subalgebra , 2007, 0706.3623.

[3]  D. Rudolph.  The second centralizer of a Bernoulli shift is just its powers , 1978 .

[4]  Type II$_1$ factors with arbitrary countable endomorphism group , 2013, 1301.2618.

[5]  U. Haagerup,et al.  Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one , 1989 .

[6]  Rodney G. Downey,et al.  THE ISOMORPHISM PROBLEM FOR TORSION-FREE ABELIAN GROUPS IS ANALYTIC COMPLETE. , 2008 .

[7]  P. Porcelli,et al.  On rings of operators , 1967 .

[8]  N. Ozawa.  A comment on free group factors , 2010 .

[9]  Rüdiger Göbel,et al.  Automorphism groups of fields , 1994 .

[10]  K. Dykema,et al.  Compressions of free products of von Neumann algebras , 1999, math/9911011.

[11]  S. Vaes,et al.  Every compact group arises as the outer automorphism group of a II_1 factor , 2007, 0705.1420.

[12]  S. Vaes.  Explicit computations of all finite index bimodules for a family of II$_1$ factors , 2007, 0707.1458.

[13]  S. Popa,et al.  A CLASS OF SUPERRIGID GROUP VON NEUMANN ALGEBRAS , 2010, 1007.1412.

[14]  Theworkof Alain Connes.  CLASSIFICATION OF INJECTIVE FACTORS , 1981 .

[15]  A. O. Houcine.  On hyperbolic groups , 2006 .

[16]  von Neumann algebras.  Solid Von Neumann Algebras , 2003 .

[17]  On the superrigidity of malleable actions with spectral gap , 2006, math/0608429.

[18]  Cyril Houdayer,et al.  Bass-Serre rigidity results in von Neumann algebras , 2008, 0805.1566.

[19]  S. Popa,et al.  Unique Cartan decomposition for II1 factors arising from arbitrary actions of free groups , 2011, 1111.6951.

[20]  J. Schwartz.  Two finite, non-hyperfinite, non-isomorphic factors , 1963 .

[21]  S. Popa.  SOME COMPUTATIONS OF 1-COHOMOLOGY GROUPS AND CONSTRUCTION OF NON-ORBIT-EQUIVALENT ACTIONS , 2004, Journal of the Institute of Mathematics of Jussieu.

[22]  There is no separable universal II_1-factor , 2002, math/0210411.

[23]  F. Richman.  The constructive theory of countable abelian $p$-groups. , 1973 .

[24]  Strong rigidity of II1 factors arising from malleable actions of w-rigid groups, I , 2003, math/0305306.

[25]  Strong rigidity of generalized Bernoulli actions and computations of their symmetry groups , 2006, math/0605456.

[26]  S. Popa,et al.  Unique Cartan decomposition for II_1 factors arising from arbitrary actions of hyperbolic groups , 2012, 1201.2824.

[27]  S. Deprez.  Explicit examples of equivalence relations and II₁ factors with prescribed fundamental group and outer automorphism group , 2010, 1010.3612.

[28]  A. Connes,et al.  Property T for von Neumann Algebras , 1985 .

[29]  A. Ioana.  W*-superrigidity for Bernoulli actions of property (T) groups , 2010, 1002.4595.

[30]  Nathanial P. Brown Narutaka Ozawa.  C*-Algebras and Finite-Dimensional Approximations , 2008 .

[31]  S. Popa,et al.  Group measure space decomposition of II1 factors and W*-superrigidity , 2009, 0906.2765.

[32]  A Kurosh-type theorem for type II1 factors , 2004, math/0401121.

[33]  S. Popa,et al.  Amalgamated free products of weakly rigid factors and calculation of their symmetry groups , 2005, math/0505589.

[34]  Y. Ueda,et al.  Rigidity of free product von Neumann algebras , 2015, Compositio Mathematica.