## Acta Mathematica

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

#### 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.

#### Article information

Source
Acta Math., Volume 200, Number 1 (2008), 85-153.

Dates
Revised: 20 November 2007
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.acta/1485891958

Digital Object Identifier
doi:10.1007/s11511-008-0024-5

Mathematical Reviews number (MathSciNet)
MR2386109

Zentralblatt MATH identifier
1149.46047

Rights

#### Citation

Ioana, Adrian; Peterson, Jesse; Popa, Sorin. Amalgamated free products of weakly rigid factors and calculation of their symmetry groups. Acta Math. 200 (2008), no. 1, 85--153. doi:10.1007/s11511-008-0024-5. https://projecteuclid.org/euclid.acta/1485891958

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