Abstract
We consider amalgamated free product II1 factors M = M1*BM2*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 Mi’s. We apply this to the case where the Mi’s are w-rigid II1 factors, with B equal to either C, to a Cartan subalgebra A in Mi, or to a regular hyperfinite II1 subfactor R in Mi, to obtain the following type of unique decomposition results, àla Bass–Serre: If M = (N1 * CN2*C…)t, for some t > 0 and some other similar inclusions of algebras C ⊂ Ni then, after a permutation of indices, (B ⊂ Mi) is inner conjugate to (C ⊂ Ni)t, for all i. Taking B = C and $ M_{i} = {\left( {L{\left( {Z^{2} \rtimes F_{2} } \right)}} \right)}^{{t_{i} }} $, with {ti}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.
Citation
Adrian Ioana. Jesse Peterson. Sorin Popa. "Amalgamated free products of weakly rigid factors and calculation of their symmetry groups." Acta Math. 200 (1) 85 - 153, 2008. https://doi.org/10.1007/s11511-008-0024-5
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