Unique prime decomposition results for factors coming from wreath product groups

Sizemore, James Owen; Winchester, Adam

doi:10.2140/pjm.2013.265.221

Cited by 12 publications

(4 citation statements)

References 27 publications

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“…, M m are factors associated with free quantum groups, the tensor product factor M 1 ⊗ • • • ⊗ M m retains the integer m and each factor M i up to stable isomorphism, after permutation of the indices. For other UPF results in the framework of II 1 factors, we refer the reader to [CKP14,CSU11,Pe06,Sa09,SW11].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Unique prime factorization and bicentralizer problem for a class of type III factors

Houdayer

Isono

2017

Advances in Mathematics

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Abstract. We show that whenever m ≥ 1 and M1, . . . , Mm are nonamenable factors in a large class of von Neumann algebras that we call C (AO) and which contains all free Araki-Woods factors, the tensor product factor M1 ⊗· · ·⊗Mm retains the integer m and each factor Mi up to stable isomorphism, after permutation of the indices. Our approach unifies the Unique Prime Factorization (UPF) results from [OP03,Is14] and moreover provides new UPF results in the case when M1, . . . , Mm are free Araki-Woods factors. In order to obtain the aforementioned UPF results, we show that Connes's bicentralizer problem has a positive solution for all type III1 factors in the class C (AO) .

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Unique prime factorization and bicentralizer problem for a class of type III factors

Houdayer

Isono

2017

Advances in Mathematics

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“…Popa's pioneering work [Pop06b, Pop06c] allowed one to distinguish between the group von Neumann algebras of , as is an infinite property (T) group, while Ioana, Popa, and Vaes used a wreath product construction to obtain the first class of groups that are entirely remembered by their von Neumann algebras [IPV13]. Subsequently, several other rigidity results have been obtained for von Neumann algebras of wreath products including primeness, relative solidity, and product rigidity, see [Ioa07, Pop08, CI10, Ioa11, IPV13, SW13, CPS12, BV14, IM19, Dri21, CDD21]. Theorem A establishes a new general rigidity result for wreath product groups by showing that products of arbitrary non-amenable wreath product groups with amenable base satisfy an analogue of Monod and Shalom's unique prime factorization result.…”

Section: Introductionmentioning

confidence: 99%

Measure equivalence rigidity via s-malleable deformations

Drimbe¹

2023

Compositio Math.

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We single out a large class of groups ${\rm {\boldsymbol {\mathscr {M}}}}$ for which the following unique prime factorization result holds: if $\Gamma _1,\ldots,\Gamma _n\in {\rm {\boldsymbol {\mathscr {M}}}}$ and $\Gamma _1\times \cdots \times \Gamma _n$ is measure equivalent to a product $\Lambda _1\times \cdots \times \Lambda _m$ of infinite icc groups, then $n \ge m$ , and if $n = m$ , then, after permutation of the indices, $\Gamma _i$ is measure equivalent to $\Lambda _i$ , for all $1\leq i\leq n$ . This provides an analogue of Monod and Shalom's theorem [Orbit equivalence rigidity and bounded cohomology, Ann. of Math. 164 (2006), 825–878] for groups that belong to ${\rm {\boldsymbol {\mathscr {M}}}}$ . Class ${\rm {\boldsymbol {\mathscr {M}}}}$ is constructed using groups whose von Neumann algebras admit an s-malleable deformation in the sense of Sorin Popa and it contains all icc non-amenable groups $\Gamma$ for which either (i) $\Gamma$ is an arbitrary wreath product group with amenable base or (ii) $\Gamma$ admits an unbounded 1-cocycle into its left regular representation. Consequently, we derive several orbit equivalence rigidity results for actions of product groups that belong to ${\rm {\boldsymbol {\mathscr {M}}}}$ . Finally, for groups $\Gamma$ satisfying condition (ii), we show that all embeddings of group von Neumann algebras of non-amenable inner amenable groups into $L(\Gamma )$ are ‘rigid’. In particular, we provide an alternative solution to a question of Popa that was recently answered by Ding, Kunnawalkam Elayavalli, and Peterson [Properly Proximal von Neumann Algebras, Preprint (2022), arXiv:2204.00517].

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“…Here, by a noncommutative Bernoulli shift we always mean an action of the form Γ (B 0 , ϕ 0 ) ⊗Γ where B 0 is a non-trivial von Neumann algebra with separable predual, ϕ 0 is a faithful normal state on B 0 and Γ is a countable group acting by shifting the tensor components. It is known that if Γ is non-amenable and B 0 is amenable, then the crossed product (B 0 , ϕ 0 ) ⊗Γ is prime [Po06b,SW11,Ma16]. By exploiting the fullness of (B 0 , ϕ 0 ) ⊗Γ ⋊ Γ (see [VV14]), we are able to remove the amenability assumption on B 0 .…”

Section: Introductionmentioning

confidence: 99%

Tensor product decompositions and rigidity of full factors

Isono¹,

Marrakchi²

2019

Preprint

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We obtain several rigidity results regarding tensor product decompositions of factors. First, we show that any full factor with separable predual has at most countably many tensor product decompositions up to stable unitary conjugacy. We use this to show that the class of separable full factors with countable fundamental group is stable under tensor products. Next, we obtain new primeness and unique prime factorization results for crossed products comming from compact actions of higher rank lattices (e.g. SL(n, Z), n ≥ 3) and noncommutative Bernoulli shifts with arbitrary base (not necessarily amenable). Finally, we provide examples of full factors without any prime factorization.

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Unique prime decomposition results for factors coming from wreath product groups

Cited by 12 publications

References 27 publications

Unique prime factorization and bicentralizer problem for a class of type III factors

Unique prime factorization and bicentralizer problem for a class of type III factors

Measure equivalence rigidity via s-malleable deformations

Tensor product decompositions and rigidity of full factors

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