In von Neumann algebras, the Connes embedding problem or conjecture, due to Alain Connes, asks whether every type II1 factor on a separable Hilbert space can be embedded into the ultrapower of the hyperfinite type II1 factor by a free ultrafilter. The problem admits a number of equivalent formulations.
Let
ω
be a free ultrafilter on the natural numbers and let R be the hyperfinite type II1 factor with trace
τ
. One can construct the ultrapower
R
ω
as follows: let
l
∞
(
R
)
=
{
(
x
n
)
n
⊆
R
:
s
u
p
n
|
|
x
n
|
|
<
∞
}
be the von Neumann algebra of norm-bounded sequences and let
I
ω
=
{
(
x
n
)
∈
l
∞
(
R
)
:
l
i
m
n
→
ω
τ
(
x
n
∗
x
n
)
1
2
=
0
}
. The quotient
l
∞
(
R
)
/
I
ω
turns out to be a II1 factor with trace
τ
R
ω
(
x
)
=
l
i
m
n
→
ω
τ
(
x
n
+
I
ω
)
, where
(
x
n
)
n
is any representative sequence of
x
.
Connes' Embedding Conjecture asks whether every type II1 factor on a separable Hilbert space can be embedded into some
R
ω
.
The isomorphism class of
R
ω
is independent of the ultrafilter if and only if the continuum hypothesis is true (Ge-Hadwin and Farah-Hart-Sherman), but such an embedding property does not depend on the ultrafilter because von Neumann algebras acting on separable Hilbert spaces are, roughly speaking, very small.
