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2008 Amalgamated free products of weakly rigid factors and calculation of their symmetry groups
Adrian Ioana, Jesse Peterson, Sorin Popa
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Acta Math. 200(1): 85-153 (2008). DOI: 10.1007/s11511-008-0024-5

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.

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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

Information

Received: 28 February 2006; Revised: 20 November 2007; Published: 2008
First available in Project Euclid: 31 January 2017

zbMATH: 1149.46047
MathSciNet: MR2386109
Digital Object Identifier: 10.1007/s11511-008-0024-5

Rights: 2008 © Institut Mittag-Leffler

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