We prove some unique factorization results for tensor products of free quantum group factors.
They are type III analogues of factorization results for direct products of bi-exact groups established ...by Ozawa and Popa.
In the proof, we first take continuous cores of the tensor products, which satisfy a condition similar to condition (AO), and discuss some factorization properties for the continuous cores.
We then deduce factorization properties for the original type III factors.
We also prove some unique factorization results for crossed product von Neumann algebras by direct products of bi-exact groups.
Let
$M$
be a
$\text{II}_{1}$
factor and let
${\mathcal{F}}(M)$
denote the fundamental group of
$M$
. In this article, we study the following property of
$M$
: for any
$\text{II}_{1}$
factor
$B$
, we ...have
${\mathcal{F}}(M\,\overline{\otimes }\,B)={\mathcal{F}}(M){\mathcal{F}}(B)$
. We prove that for any subgroup
$G\leqslant \mathbb{R}_{+}^{\ast }$
which is realized as a fundamental group of a
$\text{II}_{1}$
factor, there exists a
$\text{II}_{1}$
factor
$M$
which satisfies this property and whose fundamental group is
$G$
. Using this, we deduce that if
$G,H\leqslant \mathbb{R}_{+}^{\ast }$
are realized as fundamental groups of
$\text{II}_{1}$
factors, then so are groups
$G\cdot H$
and
$G\cap H$
.
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).
The classical Gaussian functor associates to every orthogonal representation of a locally compact group
G
a probability measure preserving action of
G
called a Gaussian action. In this paper, we ...generalize this construction by associating to every affine isometric action of
G
on a Hilbert space, a one-parameter family of nonsingular Gaussian actions whose ergodic properties are related in a very subtle way to the geometry of the original action. We show that these nonsingular Gaussian actions exhibit a phase transition phenomenon and we relate it to new quantitative invariants for affine isometric actions. We use the Patterson-Sullivan theory as well as Lyons-Pemantle work on tree-indexed random walks in order to give a precise description of this phase transition for affine isometric actions of groups acting on trees. We also show that every locally compact group without property (T) admits a nonsingular Gaussian that is free, weakly mixing and of stable type
III
1
.
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.
Let
G
be a countable discrete group and consider a nonsingular Bernoulli shift action
G
↷
∏
g
∈
G
(
{
0
,
1
}
,
μ
g
)
with two base points. We prove the first rigidity result for Bernoulli shift ...actions that are not measure preserving, by proving solidity for certain non-singular Bernoulli actions, making use of a new boundary associated with such Bernoulli actions. This generalizes solidity of measure preserving Bernoulli actions by Ozawa and Chifan–Ioana. For the proof, we use anti-symmetric Fock spaces and left creation operators to construct the boundary and therefore the assumption of having two base points is crucial.
We consider some conditions similar to Ozawa's condition (AO) and prove that if a non-injective factor satisfies such a condition and has the W*CBAP, then it has no Cartan subalgebras. As a ...corollary, we prove that II1 factors of universal orthogonal and unitary discrete quantum groups have no Cartan subalgebras. We also prove that continuous cores of type III1 factors with such a condition are semisolid as a II∞ factor.
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.
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