Amalgamated Free Products of $w$-Rigid Factors and Calculation of their Symmetry Groups

• DOI: 10.48550/arxiv.math/0505589
• Publication Date: 2005-01-01
• Authors: Adrian Ioana; Jesse Peterson; Sorin Popa
• Keywords: Cartan subalgebra

We consider amalgamated free product II$_1$ factors $M = M_1 *_B M_2 ...$ and use ``deformation/rigidity'' ``intertwining'' techniques to prove that any relatively rigid von Neumann subalgebra $Q\subset M$ can be intertwined into one of the $M_i$'s. apply this case $M_i$ are w-rigid factors, with $B$ equal either $\Bbb C$, a Cartan $A$ in $M_i$, or regular hyperfinite subfactor $R$ obtain following type unique decomposition results, à la Bass-Serre: If (N_1 *_C N_2 ...)^t$, for some $t>0$ other similar inclusions algebras $C\subset N_j$ then, after permutation indices, $(B\subset M_i)$ is inner conjugate $(C\subset N_i)^t$, $\forall i$. Taking $B=\Bbb C$ $M_i (L(\Bbb Z^2 \rtimes \Bbb F_{2}))^{t_i}$, $\{t_i\}_{i\geq 1}=S$ given countable subgroup R_+^*$, we continuously many non stably isomorphic $M$ fundamental group $\mycal F(M)$ $S$. For $B=A$, new class decomposition, large subclass satisfying F(M)=\{1\}$ Out$(M)$ abelian calculable. $B=R$, get examples F(M)=\{1\}$, Out$(M)=K$, separable compact $K$.