, Volume 200, Issue 1, pp 85-153
Date: 22 Mar 2008

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

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Abstract

We consider amalgamated free product II1 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 II1 factors, with B equal to either C, to a Cartan subalgebra A in M i , or to a regular hyperfinite II1 subfactor R in M i , to obtain the following type of unique decomposition results, àla Bass–Serre: If M = (N 1 * CN2*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.