Some prime factorization results for type II1 factors

By Nathanial P. Brown Narutaka Ozawa, Sorin Popa. 2003

114 citations recorded by the source corpus.

We prove several unique prime factorization results for tensor products of type II1 factors coming from groups that can be realized either as subgroups of hyperbolic groups or as discrete subgroups of connected Lie groups of real rank 1. In particular, we show that if $\mathcal{R}\overline{\otimes}\mathcal{L}\mathbb{F}_{r_1}\overline{\otimes}\cdot\cdot\cdot\overline{\otimes}\mathcal{L}\mathbb{F}_{r_m}$ is isomorphic to a subfactor in $\mathcal{R}\overline{\otimes}\mathcal{L}\mathbb{F}_{s_1}\overline{\otimes}\cdot\cdot\cdot\overline{\otimes}\mathcal{L}\mathbb{F}_{s_n}$, for some 2≤ri,sj≤∞, then m≤n.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c829d1f9560ce0cab6ec97483d3a642e03959890

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Mathematical Physics

Citations in the corpus

RIGIDITY FOR VON NEUMANN ALGEBRAS (2017, methodology): “In Popa [1983], Popa proved that II1 factors arising from free groups with uncountably many generators are prime and do not have Cartan subalgebras.”
Rigidity for von Neumann algebras and their invariants (2010, background): “balgebra is injective. Obviously non-hyperﬁnite solid II1 factors, as well as all their non-hyperﬁnite subfactors, are prime. A combination of techniques from [Po01, Oz03] then allowed Ozawa and Popa [OP03] to introduce a family of II1 factors that have an essentially unique tensor product decomposi”
Unique prime factorization and bicentralizer problem for a class of type III factors (2015, unknown): “This will follow from the proofs of [OP03, Proposition 12] and [Is14, Lemma 3.3.2].”
On the superrigidity of malleable actions with spectral gap (2006, background): “The results in Theorems 1.1 through 1.6 add to the plethora of rigidity phenomena involving product groups that have been discovered in recent years in group theory, OE ergodic theory, Borel equivalence relations and von Neumann algebras/II1 factors ([MoSh], [HK], [ OP ], [Mo], [BSh], etc).”
L2-rigidity in von Neumann algebras (2009, methodology): “As a consequence of the previous theorem, we obtain from [ OP ] the following unique prime factorization result.”
Ideal bicombings for hyperbolic groups and applications (2003, methodology): “The geometric counterpart of this statement (prime factorization for quasi-isometries instead of ME) follows from arguments of [9,10,20]; an analogue for von Neumann algebras has been proposed recently by Ozawa–Popa [28].”
Classification and rigidity for von Neumann algebras (2012, methodology): “nt A′ ∩L(Γ) of any diﬀuse von Neumann subalgebra A ⊂L(Γ) is amenable [Oz03]. In particular, L(Γ) and all of its non-hyperﬁnite subfactors are prime. Techniques from [Po01b, Oz03] werethen combined in [OP03] to provideafamily ofII1 factorsthat can be uniquely written as a tensor product of prime fact”
On Ozawa's Property for Free Group Factors (2006, background): “M)u ∗ = R 1/t , after some “re-scaling” with an appropriate t > 0 of the decomposition M = P ⊗R, as in ([ OP ]).”
ON THE STRUCTURAL THEORY OF II1 FACTORS OF NEGATIVELY CURVED GROUPS, II. ACTIONS BY PRODUCT GROUPS (2011, unknown): “The result then follows by appealing to Proposition 12 of [26].”
Boundary amenability of relatively hyperbolic groups (2005, background): “Hence, the relevant results in [Oz2][ OP ][Oz3] are applicable to such G. Examples of such groups include the fundamental groups of complete non-compact finite-volume Riemannian manifolds with pinched negative sectional curvature (which are hyperbolic relative to nilpotent cusp subgroups) [Fa] and l”
Rigidity results for wreath product II1 factors (2006, background): “The first part of the following result is a consequence of the proof of Proposition 12 in [ OP ].”