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

By Adrian Ioana, J. Peterson, Sorin Popa. 2005

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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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/246f0fba2b5d2e0f464f6a6647c51e5f1fdcffd1

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Citations in the corpus

On Ozawa's Property for Free Group Factors (2006, methodology): “A related deformation for free products of arbitrary factors has been used extensively in ([ IPeP ]).”
Type III factors with unique Cartan decomposition (2012, background): “given by (amalgamated) free products. In [Oz04], it was shown that the free product M= M1 ∗M2 of ﬁnite factors Micontaining a dense exact C∗-subalgebra, is always prime, unless M1 ∼= C or M2 ∼= C. In [IPP05], a systematic Bass-Serre theory for amalgamated free products M= M1 ∗BM2 of ﬁnite von Neuman”
Ergodic Subequivalence Relations Induced by a Bernoulli Action (2008, background): “This in turn implies that we can find i ∈ F such that a corner of P embeds into Γ B � Γi (by the beginning of the proof of [ IPP , Th. 4.3]).”
Cocycle Superrigidity for Profinite Actions of Property (T) Groups (2008, unknown): “ﬁes x⊥L∞Y ⋊ α Γ and LΓx⊂ P n i=1 x iLΓ, for some x 1,x 2,..,x n ∈M, then x= 0. Note that the proof of this fact is based on an argument due to S. Popa (see section 3 in [P7] and the proofs of 1.1. in [IPP] and 2.10. in [P7]). For every ε>0, let f ε(t) = 1 (ε,+∞)(t)t−1/2 and deﬁne x ε = xf ε(E LΓ(”
W*‐superrigidity for group von Neumann algebras of left–right wreath products (2012, background): “groups such that two groups G and Λ in the class C must be isomorphic whenever they have isomorphic group II1 factors; see, for example, [12, 21, 23, 26].”
Classification and rigidity for von Neumann algebras (2012, methodology): “rmalized by the unitaries vn, the whole unit ball of B must also have uniformly bounded length. It is clear that both M1 and M2 have uniformly bounded length. Conversely, the main technical result of [IPP05] shows that any subalgebra B with this property can be unitarily conjugated into either M1 or”
Group measure space decomposition of II1 factors and W*-superrigidity (2009, methodology): “balgebra A, through these techniques, one needs some amount of rigidity for one of the group actions and a deformation property for the other (like for example in 6.2 of [Po01], 7.1 of [Po04], 7.7 of [IPP05], 1.5 of [Po06a], etc). Since in W∗-superrigidity statements all assumptions must be on the s”
Approximation properties and absence of Cartan subalgebra for free Araki-Woods factors (2010, unknown): “Mp(L(R)p)′′, that is, L(R)pis regular in pMp. 5.5. Further structural results for free products. A free malleable deformation for (amalgamated) free products of von Neumann algebras was discovered in [24]. Using ideas and techniques of [6, 19, 24, 27] and of the present paper, we obtain the followin”
On cocycle superrigidity for Gaussian actions (2009, background): “) Their deformation was inspired by the free product deformation used in [20].”
Orbit Equivalence and Measured Group Theory (2010, background): “This property (never satisﬁed by standard Bernoulli shifts) entails several rigidity phenomena (see for instance [Pop06a, IPP08, GP05]).”
Strong solidity of group factors from lattices in SO(n,1) and SU(n,1) (2010, background): “3 in [11] shows that this is the only condition necessary for the theorem to obtain.”
Independence properties in subalgebras of ultraproduct II$_1$ factors (2013, unknown): “s, Proc. Amer. Math. Soc. 7(1956), 2326. [FM] J. Feldman, C.C. Moore: Ergodic equivalence relations, cohomology, and von Neumann algebras I, II, Trans. Amer. Math. Soc. 234 (1977), 289-324, 325- 359. [IPP] A. Ioana, J. Peterson, S. Popa: Amalgamated Free Products of w-Rigid Factors and Calculation o”
A SURVEY OF MEASURED GROUP THEORY (2009, background): “This strategy can be carried out in other cases of very rigid actions as in [62,75,88,96].”
ON THE STRUCTURAL THEORY OF II1 FACTORS OF NEGATIVELY CURVED GROUPS, II. ACTIONS BY PRODUCT GROUPS (2011, background): “4 in [13], one can find unitaries ui ∈ U(N) such that (5.”
Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups (2005, methodology): “be used to obtain an OE rigidity result of Bass-Serre type, in the spirit of results in ([ IPeP ]).”
L2-rigidity in von Neumann algebras (2009, methodology): “M2. We will show in this section that this is indeed the case, we do this by first showing that Q must be rigid with respect to the deformations used in [ IPP ], then we may apply the word reduction argument in [IPP] (Theorem 4.3) which gives the result.”
A CLASS OF SUPERRIGID GROUP VON NEUMANN ALGEBRAS (2010, methodology): “ is a non-amenable factor and Qis a diﬀuse factor, then the II1 factors Mn are non-isomorphic. When P and Qare semi-exact, this follows from [Oz04, Corollary 3.5]. In the general case, the methods of [IPP05] can be used, see [Pe06, Theorem 1.4] and [CH08, Theorem 1.1]. 36 ADRIAN IOANA, SORIN POPA, A”