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

Abstract

We consider amalgamated free product II₁ factors M = M_1*BM_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 * CN_2*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.