Amalgamated free products of weakly rigid factors and calculation of their symmetry groups

Adrian Ioana Department of Mathematics, University of California, Los Angeles Jesse Peterson Department of Mathematics, University of California, Los Angeles Sorin Popa Department of Mathematics, University of California, Los Angeles

TBD mathscidoc:1701.331991

Acta Mathematica, 200, (1), 85-153, 2006.2
We consider amalgamated free product II_{1}factors$M$=$M$_{1*$B$}$M$_{2*$B$}… and use “deformation/rigidity” and “intertwining” techniques to prove that any relatively rigid von Neumann subalgebra$Q$⊂$M$can be unitarily conjugated into one of the$M$_{$i$}’s. We apply this to the case where the$M$_{$i$}’s are w-rigid II_{1}factors, with$B$equal to either$C$, to a Cartan subalgebra$A$in$M$_{$i$}, or to a regular hyperfinite II_{1}subfactor$R$in$M$_{$i$}, to obtain the following type of unique decomposition results, àla Bass–Serre: If$M$= ($N$_{1 * C}N_{2*$C$}…)^{$t$}, for some$t$> 0 and some other similar inclusions of algebras$C$⊂$N$_{$i$}then, after a permutation of indices, ($B$⊂$M$_{$i$}) is inner conjugate to ($C$⊂$N$_{$i$})^{$t$}, for all$i$. Taking$B$=$C$and $ M_{i} = {\left( {L{\left( {Z^{2} \rtimes F_{2} } \right)}} \right)}^{{t_{i} }} $ , with {$t$_{$i$}}_{$i$⩾1}=$S$a given countable subgroup of$R$_{+}^{*}, we obtain continuously many non-stably isomorphic factors$M$with fundamental group $ {\user1{\mathcal{F}}}{\left( M \right)} $ equal to$S$. For$B$=$A$, we obtain a new class of factors$M$with unique Cartan subalgebra decomposition, with a large subclass satisfying $ {\user1{\mathcal{F}}}{\left( M \right)} = {\left\{ 1 \right\}} $ and Out(M) abelian and calculable. Taking$B$=$R$, we get examples of factors with $ {\user1{\mathcal{F}}}{\left( M \right)} = {\left\{ 1 \right\}} $ , Out($M$) =$K$, for any given separable compact abelian group$K$.
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@inproceedings{adrian2006amalgamated,
  title={Amalgamated free products of weakly rigid factors and calculation of their symmetry groups},
  author={Adrian Ioana, Jesse Peterson, and Sorin Popa},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203351583487700},
  booktitle={Acta Mathematica},
  volume={200},
  number={1},
  pages={85-153},
  year={2006},
}
Adrian Ioana, Jesse Peterson, and Sorin Popa. Amalgamated free products of weakly rigid factors and calculation of their symmetry groups. 2006. Vol. 200. In Acta Mathematica. pp.85-153. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203351583487700.
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