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 33,25 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).
We prove that a large class of nonamenable almost periodic type III
1
factors
M
, including all McDuff factors that tensorially absorb
R
∞
and all free Araki–Woods factors, satisfy Haagerup–Størmer’s ...conjecture (1988): any pointwise inner automorphism of
M
is the composition of an inner and a modular automorphism.
We study several model-theoretic aspects of W
∗
-probability spaces, that is,
σ
-finite von Neumann algebras equipped with a faithful normal state. We first study the existentially closed W
∗
-spaces ...and prove several structural results about such spaces, including that they are type III
1
factors that tensorially absorb the Araki–Woods factor
R
∞
. We also study the existentially closed objects in the restricted class of W
∗
-probability spaces with Kirchberg’s QWEP property, proving that
R
∞
itself is such an existentially closed space in this class. Our results about existentially closed probability spaces imply that the class of type III
1
factors forms a
∀
2
-axiomatizable class. We show that for
λ
∈
(
0
,
1
)
, the class of III
λ
factors is not
∀
2
-axiomatizable but is
∀
3
-axiomatizable; this latter result uses a version of Keisler’s Sandwich theorem adapted to continuous logic. Finally, we discuss some results around elementary equivalence of III
λ
factors. Using a result of Boutonnet, Chifan, and Ioana, we show that, for any
λ
∈
(
0
,
1
)
, there is a family of pairwise non-elementarily equivalent III
λ
factors of size continuum. While we cannot prove the same result for III
1
factors, we show that there are at least three pairwise non-elementarily equivalent III
1
factors by showing that the class of full factors is preserved under elementary equivalence.
We address the problem to characterise closed type I subgroups of the automorphism group of a tree. Even in the well-studied case of Burger–Mozes’ universal groups, non-type I criteria were unknown. ...We prove that a huge class of groups acting properly on trees are not of type I. In the case of Burger–Mozes groups, this yields a complete classification of type I groups among them. Our key novelty is the use of von Neumann algebraic techniques to prove the stronger statement that the group von Neumann algebra of the groups under consideration is non-amenable.
Abstract
We investigate factoriality, Connes' type III invariants and fullness of arbitrary amalgamated free product von Neumann algebras using Popa's deformation/rigidity theory. Among other things, ...we generalize many previous structural results on amalgamated free product von Neumann algebras and we obtain new examples of full amalgamated free product factors for which we can explicitely compute Connes' type III invariants.
Let
$I$
be any nonempty set and let
$(M_{i},\unicodeSTIX{x1D711}_{i})_{i\in I}$
be any family of nonamenable factors, endowed with arbitrary faithful normal states, that belong to a large class
...${\mathcal{C}}_{\text{anti}\text{-}\text{free}}$
of (possibly type
$\text{III}$
) von Neumann algebras including all nonprime factors, all nonfull factors and all factors possessing Cartan subalgebras. For the free product
$(M,\unicodeSTIX{x1D711})=\ast _{i\in I}(M_{i},\unicodeSTIX{x1D711}_{i})$
, we show that the free product von Neumann algebra
$M$
retains the cardinality
$|I|$
and each nonamenable factor
$M_{i}$
up to stably inner conjugacy, after permutation of the indices. Our main theorem unifies all previous Kurosh-type rigidity results for free product type
$\text{II}_{1}$
factors and is new for free product type
$\text{III}$
factors. It moreover provides new rigidity phenomena for type
$\text{III}$
factors.
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