On Ergodic Embeddings of Factors Popa, Sorin
Communications in mathematical physics,
06/2021, Volume:
384, Issue:
2
Journal Article
Peer reviewed
Open access
An inclusion of von Neumann factors
M
⊂
M
is
ergodic
if it satisfies the irreducibility condition
M
′
∩
M
=
C
. We investigate the relation between this and several stronger ergodicity properties, ...such as
R
-
ergodicity
, which requires
M
to admit an embedding of the hyperfinite II
1
factor
R
↪
M
that’s ergodic in
M
. We prove that if
M
is
continuous
(i.e., non type I) and contains a maximal abelian
∗
-subalgebra of
M
, then
M
⊂
M
is
R
-ergodic. This shows in particular that any continuous factor contains an ergodic copy of
R
.
We consider crossed product II^sub 1^ factors (ProQuest: Formulae and/or non-USASCII text omitted; see image) , with G discrete ICC groups that contain infinite normal subgroups with the relative ...property (T) and σ trace preserving actions of G on finite von Neumann algebras N that are "malleable" and mixing. Examples are the actions of G by Bernoulli shifts (classical and non-classical) and by Bogoliubov shifts. We prove a rigidity result for isomorphisms of such factors, showing the uniqueness, up to unitary conjugacy, of the position of the group von Neumann algebra L(G) inside M. We use this result to calculate the fundamental group of M, (ProQuest: Formulae and/or non-USASCII text omitted; see image) , in terms of the weights of the shift σ, for (ProQuest: Formulae and/or non-USASCII text omitted; see image) and other special arithmetic groups. We deduce that for any subgroup S^sub +^^sup *^ there exist II^sub 1^ factors M (separable if S is countable or S=^sub +^^sup *^) with (ProQuest: Formulae and/or non-USASCII text omitted; see image) . This brings new light to a long standing open problem of Murray and von Neumann. PUBLICATION ABSTRACT
We show that the Kadison–Singer problem, asking whether the pure states of the diagonal subalgebra
ℓ
∞
N
⊂
B
(
ℓ
2
N
)
have unique state extensions to
B
(
ℓ
2
N
)
, is equivalent to a similar ...statement in II
1
factor framework, concerning the ultrapower inclusion
D
ω
⊂
R
ω
, where
D
is the
Cartan subalgebra
of the hyperfinite II
1
factor
R
(i.e., a maximal abelian *-subalgebra of
R
whose normalizer generates
R
, e.g.
D
=
L
∞
(
0
,
1
Z
)
⊂
L
∞
(
0
,
1
Z
⋊
Z
=
R
)
, and
ω
is a free ultrafilter. Instead, we prove here that if
A
is any
singular
maximal abelian *-subalgebra of
R
(i.e., whose normalizer consists of the unitary group of
A
, e.g.
A
=
L
(
Z
)
⊂
L
∞
(
0
,
1
Z
)
⋊
Z
=
R
), then the inclusion
A
ω
⊂
R
ω
does satisfy the Kadison–Singer property.
We prove that for any free ergodic probability measure-preserving action
F
n
↷
(
X
,
μ
)
of a free group on
n
generators
F
n
,
2
≤
n
≤
∞
, the associated group measure space II
1
factor
L
∞
(
X
)
⋊
F
...n
has
L
∞
(
X
) as its unique Cartan subalgebra, up to unitary conjugacy. We deduce that group measure space II
1
factors arising from actions of free groups with different number of generators are never isomorphic. We actually prove unique Cartan decomposition results for II
1
factors arising from arbitrary actions of a much larger family of groups, including all free products of amenable groups and their direct products.
We prove that any isomorphism theta:M^sub 0^M of group measure space II^sub 1^ factors, (ProQuest: Formulae and/or non-USASCII text omitted; see image) , (ProQuest: Formulae and/or non-USASCII text ...omitted; see image) , with G^sub 0^ an ICC group containing an infinite normal subgroup with the relative property (T) of Kazhdan-Margulis (i.e. G^sub 0^w-rigid) and σ a Bernoulli action of some ICC group G, essentially comes from an isomorphism of probability spaces which conjugates the actions with respect to some identification G^sub 0^G. Moreover, any isomorphism theta of M^sub 0^ onto a "corner" pMp of M, for pM an idempotent, forces p=1. In particular, all group measure space factors associated with Bernoulli actions of w-rigid ICC groups have trivial fundamental group and any isomorphism of such factors comes from an isomorphism of the corresponding groups. This settles a "group measure space version" of Connes rigidity conjecture, shown in fact to hold true in a greater generality than just for ICC property (T) groups. We apply these results to ergodic theory, establishing new strong rigidity and superrigidity results for orbit equivalence relations. PUBLICATION ABSTRACT
We undertake a systematic study of W
∗
-rigidity paradigms for the embeddability relation
↪
between separable
II
1
factors and its stable version
↪
s
, obtaining large families of non stably ...isomorphic
II
1
factors that are mutually embeddable and families of
II
1
factors that are mutually non stably embeddable. We provide an augmentation functor
G
↦
H
G
from the category of groups into icc groups, so that
L
(
H
G
1
)
↪
s
L
(
H
G
2
)
iff
G
1
↪
G
2
. We construct complete intervals of
II
1
factors, including a strict chain of
II
1
factors
(
M
k
)
k
∈
Z
with the property that if
N
is any
II
1
factor with
M
i
↪
s
N
and
N
↪
s
M
j
, then
N
≅
M
k
t
for some
i
≤
k
≤
j
and
t
>
0
.
We prove that if
A
is a non-separable abelian tracial von Neuman algebra then its free powers
A
∗
n
,2≤
n
≤∞, are mutually non-isomorphic and with trivial fundamental group,
, whenever 2≤
n
<∞. This ...settles the non-separable version of the free group factor problem.
Percutaneous procedures to divert blood flow from one blood vessel to another can be performed with intravascular catheters but demand a method to align a crossing needle from one vessel to another. ...Fluoroscopic imaging alone is not adequate, and it is preferable to have a sensor on one catheter that detects the correct alignment of an incoming needle. This can be implemented by generating dipole electric fields from the crossing catheter which are detected by a receiving catheter in the target vessel and, thus, can calculate and display the degree of alignment, permitting the operator to rotate the crossing catheter to guarantee alignment when deploying a crossing needle. Catheters were built using this concept and evaluated in vitro. The results show that accurate alignment is achieved, and a successful crossing can be made. The concept is being further developed for further clinical evaluation.
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