Amalgamated free products of weakly rigid factors and calculation of their symmetry groups |
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Authors: | Adrian Ioana Jesse Peterson Sorin Popa |
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Institution: | (1) Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095-155505, USA |
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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 , with {t
i
}
i⩾1 = S a given countable subgroup of R
+
*, we obtain continuously many non-stably isomorphic factors M with fundamental group equal to S. For B = A, we obtain a new class of factors M with unique Cartan subalgebra decomposition, with a large subclass satisfying and Out(M) abelian and calculable. Taking B = R, we get examples of factors with , Out(M) = K, for any given separable compact abelian group K. |
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Keywords: | |
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