Amalgamated Free Products of $w$Rigid Factors and Calculation of their Symmetry Groups
Abstract
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 $Q\subset M$ can be intertwined into one of the $M_i$'s. We apply this to the case $M_i$ are wrigid II$_1$ factors, with $B$ equal to either $\Bbb C$, 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, à la BassSerre: If $M = (N_1 *_C N_2 *_C ...)^t$, for some $t>0$ and some other similar inclusions of algebras $C\subset N_j$ then, after a permutation of indices, $(B\subset M_i)$ is inner conjugate to $(C\subset N_i)^t$, $\forall i$. Taking $B=\Bbb C$ and $M_i = (L(\Bbb Z^2 \rtimes \Bbb F_{2}))^{t_i}$, with $\{t_i\}_{i\geq 1}=S$ a given countable subgroup of $\Bbb R_+^*$, we obtain continuously many non stably isomorphic factors $M$ with fundamental group $\mycal F(M)$ equal to $S$. For $B=A$, we obtain a new class of factors $M$ with unique Cartan subalgebra decomposition, with a large subclass satisfying $\mycal F(M)=\{1\}$ and Out$(M)$ abelian and calculable. Taking $B=R$, we get examples of factors with $\mycal F(M)=\{1\}$, Out$(M)=K$, for any given separable compact abelian group $K$.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 2005
 DOI:
 10.48550/arXiv.math/0505589
 arXiv:
 arXiv:math/0505589
 Bibcode:
 2005math......5589I
 Keywords:

 Mathematics  Operator Algebras;
 Mathematics  Group Theory;
 46L55;
 46L10;
 46L40;
 22D25;
 22D40;
 28D15
 EPrint:
 final version, to appear as such in Acta Mathematica