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Amalgamated Free Products of $w$-Rigid Factors and Calculation of their Symmetry Groups | | A. Ioana
; J. Peterson
; S. Popa
; | | Date: |
27 May 2005 | | Subject: | Operator Algebras; Group Theory MSC-class: 46L55, 46L10, 46L40, 22D25, 22D40, 28D15 | math.OA math.GR | | 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 $Qsubset M$ can be intertwined into one of the $M_i$’s. We apply this to the case $M_i$ are w-rigid 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 Bass-Serre: If $M = (N_1 *_C N_2 *_C ...)^t$, for some $t>0$ and some other similar inclusions of algebras $Csubset N_j$ then, after a permutation of indices, $(Bsubset M_i)$ is inner conjugate to $(Csubset N_i)^t$, $forall i$. Taking $B=Bbb C$ and $M_i = (L(Bbb Z^2
times Bbb F_{2}))^{t_i}$, with ${t_i}_{igeq 1}=S$ 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 $K$. | | Source: | arXiv, math.OA/0505589 | | Services: | Forum | Review | PDF | Favorites | |
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