STRUCTURE AND RANDOMNESS IN II1 FACTORS
Shanghai, April 1st (US time) 2017
Sorin Popa
1/13
STRUCTURE AND RANDOMNESS IN II1 FACTORS
Shanghai, April 1st (US time) 2017
Sorin Popa
HAPPY 70th ANNIVERSARY ALAIN !!
1/13
Constructing II1 factors from various data
• Γ (countable discrete) group 7−→ group von Neumann algebra L(Γ). It is
a II1 factor iff Γ is ICC (e.g. Z o Zk , k ≥ 1; Fn , PSL(n, Z), n ≥ 2).
2/13
Constructing II1 factors from various data
• Γ (countable discrete) group 7−→ group von Neumann algebra L(Γ). It is
a II1 factor iff Γ is ICC (e.g. Z o Zk , k ≥ 1; Fn , PSL(n, Z), n ≥ 2).
• Γ y X measure preserving (m.p.) action of Γ on the probability space
(X , µ) 7−→ group measure space vN algebra L∞ (X ) o Γ. It is a II1 factor if
Γ y X is free ergodic, in which case A = L∞ (X ) is maximal abelian in
L∞ (X ) o Γ and its normalizer generates L∞ (X ) o Γ, i.e. A is a Cartan
subalgebra.
2/13
Constructing II1 factors from various data
• Γ (countable discrete) group 7−→ group von Neumann algebra L(Γ). It is
a II1 factor iff Γ is ICC (e.g. Z o Zk , k ≥ 1; Fn , PSL(n, Z), n ≥ 2).
• Γ y X measure preserving (m.p.) action of Γ on the probability space
(X , µ) 7−→ group measure space vN algebra L∞ (X ) o Γ. It is a II1 factor if
Γ y X is free ergodic, in which case A = L∞ (X ) is maximal abelian in
L∞ (X ) o Γ and its normalizer generates L∞ (X ) o Γ, i.e. A is a Cartan
subalgebra.
• Γ y X m.p. action 7−→ (countable) equivalence relation (or groupoid)
RΓ = {(t, gt) | t ∈ X }, implemented by Γ y X . This in turn gives rise to
a groupoid von Neumann algebra L(RΓ ). II1 factor iff action is ergodic
and X non-atomic. Note: L∞ (X ) always Cartan in L(RΓ ). If Γ y X free
then L(RΓ ) coincides with L∞ (X ) o Γ.
2/13
More II1 factors from operations
Using the above vN algebras as “building blocks”, one can obtain more II1
factors by using operations:
3/13
More II1 factors from operations
Using the above vN algebras as “building blocks”, one can obtain more II1
factors by using operations:
• Amplifications: M 7−→ M t , t > 0.
3/13
More II1 factors from operations
Using the above vN algebras as “building blocks”, one can obtain more II1
factors by using operations:
• Amplifications: M 7−→ M t , t > 0.
• Tensor product: (Mi )i∈I 7−→ ⊗i Mi . It is II1 factor iff all Mi are finite
factors 6= C and |I | = ∞, or at least one Mi is II1 . ⊗n (M2×2 (C))n called
the hyperfinite II1 factor, denoted R.
3/13
More II1 factors from operations
Using the above vN algebras as “building blocks”, one can obtain more II1
factors by using operations:
• Amplifications: M 7−→ M t , t > 0.
• Tensor product: (Mi )i∈I 7−→ ⊗i Mi . It is II1 factor iff all Mi are finite
factors 6= C and |I | = ∞, or at least one Mi is II1 . ⊗n (M2×2 (C))n called
the hyperfinite II1 factor, denoted R.
• Free product: (M1 , M2 ) 7−→ M1 ∗ M2 . Also, if B ⊂ Mi common vN
subalgebra, then M1 ∗B M2 is the Free product with amalgamation over B.
In general it is II1 factor....
3/13
More II1 factors from operations
Using the above vN algebras as “building blocks”, one can obtain more II1
factors by using operations:
• Amplifications: M 7−→ M t , t > 0.
• Tensor product: (Mi )i∈I 7−→ ⊗i Mi . It is II1 factor iff all Mi are finite
factors 6= C and |I | = ∞, or at least one Mi is II1 . ⊗n (M2×2 (C))n called
the hyperfinite II1 factor, denoted R.
• Free product: (M1 , M2 ) 7−→ M1 ∗ M2 . Also, if B ⊂ Mi common vN
subalgebra, then M1 ∗B M2 is the Free product with amalgamation over B.
In general it is II1 factor....
• Crossed product: (B, τ ) vN algebra with a trace (e.g. B = L∞ (X ) or
B = R), Γ y B a trace preserving action 7−→ B o Γ.
3/13
More II1 factors from operations
Using the above vN algebras as “building blocks”, one can obtain more II1
factors by using operations:
• Amplifications: M 7−→ M t , t > 0.
• Tensor product: (Mi )i∈I 7−→ ⊗i Mi . It is II1 factor iff all Mi are finite
factors 6= C and |I | = ∞, or at least one Mi is II1 . ⊗n (M2×2 (C))n called
the hyperfinite II1 factor, denoted R.
• Free product: (M1 , M2 ) 7−→ M1 ∗ M2 . Also, if B ⊂ Mi common vN
subalgebra, then M1 ∗B M2 is the Free product with amalgamation over B.
In general it is II1 factor....
• Crossed product: (B, τ ) vN algebra with a trace (e.g. B = L∞ (X ) or
B = R), Γ y B a trace preserving action 7−→ B o Γ.
• Ultraproduct of finite factors Πω Mn , notably the case Πω Mn×n (C) and
the ultrapower R ω of R (i.e., the case Mn = R, ∀n)
3/13
Classic vN algebra questions
• How do L(Γ) and L∞ (X ) o Γ depend on the group Γ and the action
Γ y X ? Especially interesting for “classic groups” (e.g., linear)
4/13
Classic vN algebra questions
• How do L(Γ) and L∞ (X ) o Γ depend on the group Γ and the action
Γ y X ? Especially interesting for “classic groups” (e.g., linear)
• More generally: How much of the data D involved in constructing
M = M(D) (groups, group actions, operations, etc) can be recovered from
the isomorphism class of M? To what extent the properties of D go
through (“survive”) to M(D)?
4/13
Classic vN algebra questions
• How do L(Γ) and L∞ (X ) o Γ depend on the group Γ and the action
Γ y X ? Especially interesting for “classic groups” (e.g., linear)
• More generally: How much of the data D involved in constructing
M = M(D) (groups, group actions, operations, etc) can be recovered from
the isomorphism class of M? To what extent the properties of D go
through (“survive”) to M(D)?
• Calculate symmetry grps of M: the fundamental group F(M)
= {t > 0 | M t ' M}, the outer automorphism group Out(M) = Aut/Int.
4/13
Classic vN algebra questions
• How do L(Γ) and L∞ (X ) o Γ depend on the group Γ and the action
Γ y X ? Especially interesting for “classic groups” (e.g., linear)
• More generally: How much of the data D involved in constructing
M = M(D) (groups, group actions, operations, etc) can be recovered from
the isomorphism class of M? To what extent the properties of D go
through (“survive”) to M(D)?
• Calculate symmetry grps of M: the fundamental group F(M)
= {t > 0 | M t ' M}, the outer automorphism group Out(M) = Aut/Int.
• Embeddibility problems.
4/13
Classic vN algebra questions
• How do L(Γ) and L∞ (X ) o Γ depend on the group Γ and the action
Γ y X ? Especially interesting for “classic groups” (e.g., linear)
• More generally: How much of the data D involved in constructing
M = M(D) (groups, group actions, operations, etc) can be recovered from
the isomorphism class of M? To what extent the properties of D go
through (“survive”) to M(D)?
• Calculate symmetry grps of M: the fundamental group F(M)
= {t > 0 | M t ' M}, the outer automorphism group Out(M) = Aut/Int.
• Embeddibility problems.
• All can be viewed as rigidity questions. Results about recovering building
data D (groups, group actions, operations, etc) of a II1 factor M = M(D)
from the isomorphism class of M will be called W ∗ -rigidity.
4/13
Lack of W ∗ -rigidity in the amenable case
• Murray-von Neumann ’43: all approximately finite dimensional (AFD) II1
factors are isomorphic to the hyperfinite factor R = M2×2 (C)⊗∞ and
F(R) = R>0 . Also, R embeds into any II1 factor M.
5/13
Lack of W ∗ -rigidity in the amenable case
• Murray-von Neumann ’43: all approximately finite dimensional (AFD) II1
factors are isomorphic to the hyperfinite factor R = M2×2 (C)⊗∞ and
F(R) = R>0 . Also, R embeds into any II1 factor M.
• Connes ’76: all amenable II1 factors are isomorphic to R. In particular,
all II1 factors M(D) arising from “amenable data” D (such as amenable
groups Γ, their actions on AFD algebras Γ y B, etc) are isomorphic to R.
For instance, L(Γn ) ' R, for Γn = Z o Zn , ∀n. Conversely, if M(D) ' R,
then D amenable. So the functor D 7→ M(D) forgets everything but the
amenability.
5/13
Lack of W ∗ -rigidity in the amenable case
• Murray-von Neumann ’43: all approximately finite dimensional (AFD) II1
factors are isomorphic to the hyperfinite factor R = M2×2 (C)⊗∞ and
F(R) = R>0 . Also, R embeds into any II1 factor M.
• Connes ’76: all amenable II1 factors are isomorphic to R. In particular,
all II1 factors M(D) arising from “amenable data” D (such as amenable
groups Γ, their actions on AFD algebras Γ y B, etc) are isomorphic to R.
For instance, L(Γn ) ' R, for Γn = Z o Zn , ∀n. Conversely, if M(D) ' R,
then D amenable. So the functor D 7→ M(D) forgets everything but the
amenability.
• Since M ,→ R implies M amenable, Connes theorem also shows that any
II1 factor M that embeds into R is isomorphic to R.
5/13
More on Connes’ 1976 paper
• Connes approximate embedding (CAE) conjecture predicts that any
(separable) II1 factor M embeds into the ultraproduct Πω Mn×n (C).
For group algebras M = L(Γ) this amounts to “simulating” Γ by fin dim
unitary groups U(n): for any finite F ⊂ Γ, m ≥ 1 and ε > 0, there exists n
and {vg }g ∈F ⊂ U(n) such that for any word w of length ≤ m in the free
group with generators in F , one has |tr (w ({vg }g ) − 1| ≤ ε if w (F ) = e
and |tr (w ({vg }g ))| ≤ ε if w (F ) 6= e.
6/13
More on Connes’ 1976 paper
• Connes approximate embedding (CAE) conjecture predicts that any
(separable) II1 factor M embeds into the ultraproduct Πω Mn×n (C).
For group algebras M = L(Γ) this amounts to “simulating” Γ by fin dim
unitary groups U(n): for any finite F ⊂ Γ, m ≥ 1 and ε > 0, there exists n
and {vg }g ∈F ⊂ U(n) such that for any word w of length ≤ m in the free
group with generators in F , one has |tr (w ({vg }g ) − 1| ≤ ε if w (F ) = e
and |tr (w ({vg }g ))| ≤ ε if w (F ) 6= e.
• A strengthening of CAE for groups requires Γ to be simulated by finite
permutation groups Sn ⊂ U(n). Such groups are called sofic. Residually
finite groups are sofic. There are no known examples of non-sofic groups!
6/13
More on Connes’ 1976 paper
• Connes approximate embedding (CAE) conjecture predicts that any
(separable) II1 factor M embeds into the ultraproduct Πω Mn×n (C).
For group algebras M = L(Γ) this amounts to “simulating” Γ by fin dim
unitary groups U(n): for any finite F ⊂ Γ, m ≥ 1 and ε > 0, there exists n
and {vg }g ∈F ⊂ U(n) such that for any word w of length ≤ m in the free
group with generators in F , one has |tr (w ({vg }g ) − 1| ≤ ε if w (F ) = e
and |tr (w ({vg }g ))| ≤ ε if w (F ) 6= e.
• A strengthening of CAE for groups requires Γ to be simulated by finite
permutation groups Sn ⊂ U(n). Such groups are called sofic. Residually
finite groups are sofic. There are no known examples of non-sofic groups!
• Connes’ proof of “M amenable =⇒ M ' R” became a major source of
inspiration in the effort to classify nuclear C ∗ -algebras (Elliott, Kirchberg,
Huaxin Lin, and most recently Tikuisis-White-Winter).
6/13
II1 factors from non-amenable data
Connes theorem shows that the functor D 7→ M(D) is taking all amenable
D into just one II1 factor M(D) = R.
7/13
II1 factors from non-amenable data
Connes theorem shows that the functor D 7→ M(D) is taking all amenable
D into just one II1 factor M(D) = R.
Examples by Connes-Jones ’82 show that even for Γ non-amenable, the
functor Γ y X 7→ L∞ (X ) o Γ can be “many to one”, making the
classification of non-amenable II1 factors M = M(D) and the calculation
of their “symmetry groups” Out(M), F(M) extremely elusive.
7/13
II1 factors from non-amenable data
Connes theorem shows that the functor D 7→ M(D) is taking all amenable
D into just one II1 factor M(D) = R.
Examples by Connes-Jones ’82 show that even for Γ non-amenable, the
functor Γ y X 7→ L∞ (X ) o Γ can be “many to one”, making the
classification of non-amenable II1 factors M = M(D) and the calculation
of their “symmetry groups” Out(M), F(M) extremely elusive.
Connes 1980: rigidity of data D entails “small” Out(M), F(M).
7/13
II1 factors from non-amenable data
Connes theorem shows that the functor D 7→ M(D) is taking all amenable
D into just one II1 factor M(D) = R.
Examples by Connes-Jones ’82 show that even for Γ non-amenable, the
functor Γ y X 7→ L∞ (X ) o Γ can be “many to one”, making the
classification of non-amenable II1 factors M = M(D) and the calculation
of their “symmetry groups” Out(M), F(M) extremely elusive.
Connes 1980: rigidity of data D entails “small” Out(M), F(M).
Much progress has been made in this direction during 2001-2016 due to
deformation-rigidity theory. This is based on the discovery that if the
data D contains the right combination of “soft and rigid” information,
then much of D can be recovered by merely knowing the isomorphism
class of the associated II1 factor M(D).
7/13
II1 factors from non-amenable data
Connes theorem shows that the functor D 7→ M(D) is taking all amenable
D into just one II1 factor M(D) = R.
Examples by Connes-Jones ’82 show that even for Γ non-amenable, the
functor Γ y X 7→ L∞ (X ) o Γ can be “many to one”, making the
classification of non-amenable II1 factors M = M(D) and the calculation
of their “symmetry groups” Out(M), F(M) extremely elusive.
Connes 1980: rigidity of data D entails “small” Out(M), F(M).
Much progress has been made in this direction during 2001-2016 due to
deformation-rigidity theory. This is based on the discovery that if the
data D contains the right combination of “soft and rigid” information,
then much of D can be recovered by merely knowing the isomorphism
class of the associated II1 factor M(D).
Deformation-rigidity techniques fit within the fundamental dichotomy
structure versus randomness, which has been the engine behind the
solution to many recent deep results in ergodic theory, graph theory,
harmonic analysis, etc.
7/13
W ∗ -rigidity for II1 factors from Bernoulli actions
• If (X0 , ν0 ) is a probability space and Γ a countable group, then Γ acts on
(X , µ) = Πg (X0 , µ0 )g = (X0 , µ0 )Γ by shifting the coordinates. Called the
Bernoulli Γ-action with base (X0 , µ0 ). It is free, ergodic (even mixing).
8/13
W ∗ -rigidity for II1 factors from Bernoulli actions
• If (X0 , ν0 ) is a probability space and Γ a countable group, then Γ acts on
(X , µ) = Πg (X0 , µ0 )g = (X0 , µ0 )Γ by shifting the coordinates. Called the
Bernoulli Γ-action with base (X0 , µ0 ). It is free, ergodic (even mixing).
Theorem (Popa 2004, 2006)
Assume Γ is either a property (T) group, or a product of a non-amenable
group with an infinite group. Let Γ y X , Λ y Y be Bernoulli actions,
where Λ is an arbitrary group. If θ : L∞ (X ) o Γ ' (L∞ (Y ) o Λ)t then
t = 1 and θ arises from a map ∆ : (X , µ) ' (Y , ν) that implements a
conjugacy of Γ y X , Λ y Y . In particular, F(L∞ (X ) o Γ)) = 1.
8/13
W ∗ -superrigidity for groups
Connes Rigidity Conjecture 1980 predicts that any property (T) ICC
group Γ is W ∗ -superrigid: if Λ is any other ICC group and
θ : L(Γ) ' L(Λ)t , then t = 1 and there exist a unitary u ∈ U(L(Λ)), an iso
δ : Γ ' Λ and γ ∈ Hom(Λ, T) such that θ = Ad(u) ◦ θγ ◦ θδ , where θγ is
the automorphism of L(Λ) implemented by γ and θδ : L(Γ) ' L(Λ) the
isomorphism implemented by δ.
9/13
W ∗ -superrigidity for groups
Connes Rigidity Conjecture 1980 predicts that any property (T) ICC
group Γ is W ∗ -superrigid: if Λ is any other ICC group and
θ : L(Γ) ' L(Λ)t , then t = 1 and there exist a unitary u ∈ U(L(Λ)), an iso
δ : Γ ' Λ and γ ∈ Hom(Λ, T) such that θ = Ad(u) ◦ θγ ◦ θδ , where θγ is
the automorphism of L(Λ) implemented by γ and θδ : L(Γ) ' L(Λ) the
isomorphism implemented by δ.
Theorem (Ioana-P-Vaes 2010)
Let Γ0 be any non-amenable group and let Γ = Γ0 o Z and consider the
natural action of Γ on I = Γ/Z. Then the (generalized wreath product)
group Λ = Z/2Z oI Γ is W∗ -superrigid.
9/13
Classification of free group measure space factors
• P ’01: If Fn , Fm ⊂ SL(2, Z), then L∞ (T2 ) o Fn ' L∞ (T2 ) o Fm implies
n = m.
10/13
Classification of free group measure space factors
• P ’01: If Fn , Fm ⊂ SL(2, Z), then L∞ (T2 ) o Fn ' L∞ (T2 ) o Fm implies
n = m.
• Ozawa-P ’07: If L∞ (X ) o Fn ' L∞ (Y ) o Fm for free ergodic profinite
Fn y X , Fm y Y , then n = m.
10/13
Classification of free group measure space factors
• P ’01: If Fn , Fm ⊂ SL(2, Z), then L∞ (T2 ) o Fn ' L∞ (T2 ) o Fm implies
n = m.
• Ozawa-P ’07: If L∞ (X ) o Fn ' L∞ (Y ) o Fm for free ergodic profinite
Fn y X , Fm y Y , then n = m.
Thm (P-Vaes ’11), incorporating results of Gaboriau ’99, Bowen ’09
1
If Fn y Y , Fm y X are arbitrary free ergodic actions for some
2 ≤ n < m ≤ ∞, then L∞ (X ) o Fn 6' L∞ (Y ) o Fm ; moreover, if
(L∞ (X ) o Fn )t ' (L∞ (Y ) o Fm )s , then (n − 1)/t = (m − 1)/s.
10/13
Classification of free group measure space factors
• P ’01: If Fn , Fm ⊂ SL(2, Z), then L∞ (T2 ) o Fn ' L∞ (T2 ) o Fm implies
n = m.
• Ozawa-P ’07: If L∞ (X ) o Fn ' L∞ (Y ) o Fm for free ergodic profinite
Fn y X , Fm y Y , then n = m.
Thm (P-Vaes ’11), incorporating results of Gaboriau ’99, Bowen ’09
1
If Fn y Y , Fm y X are arbitrary free ergodic actions for some
2 ≤ n < m ≤ ∞, then L∞ (X ) o Fn 6' L∞ (Y ) o Fm ; moreover, if
(L∞ (X ) o Fn )t ' (L∞ (Y ) o Fm )s , then (n − 1)/t = (m − 1)/s.
2
For Bernoulli actions one has:
(L∞ (X0Fn ) o Fn )t ' (L∞ (Y0Fm ) o Fm )s iff (n − 1)/t = (m − 1)/s.
In particular L(Z o Fn )t ' L(Z o Fm )s iff (n − 1)/t = (m − 1)/s.
10/13
Classification of free group measure space factors
• P ’01: If Fn , Fm ⊂ SL(2, Z), then L∞ (T2 ) o Fn ' L∞ (T2 ) o Fm implies
n = m.
• Ozawa-P ’07: If L∞ (X ) o Fn ' L∞ (Y ) o Fm for free ergodic profinite
Fn y X , Fm y Y , then n = m.
Thm (P-Vaes ’11), incorporating results of Gaboriau ’99, Bowen ’09
1
If Fn y Y , Fm y X are arbitrary free ergodic actions for some
2 ≤ n < m ≤ ∞, then L∞ (X ) o Fn 6' L∞ (Y ) o Fm ; moreover, if
(L∞ (X ) o Fn )t ' (L∞ (Y ) o Fm )s , then (n − 1)/t = (m − 1)/s.
2
For Bernoulli actions one has:
(L∞ (X0Fn ) o Fn )t ' (L∞ (Y0Fm ) o Fm )s iff (n − 1)/t = (m − 1)/s.
In particular L(Z o Fn )t ' L(Z o Fm )s iff (n − 1)/t = (m − 1)/s.
Proof uses deformation-rigidity to show unique Cartan decomposition
(up to unitary conjugacy) of the factors L∞ (X ) o Fn , thus reducing to OE
of the actions, then applying results of Gaboriau (’99), resp. Bowen (’09).
10/13
About deformation-rigidity arguments
Each one of these results is proved by using deformation rigidity
techniques. This approach exploits the tension between “soft” and “rigid”
parts of the ambient II1 factor M, to locate the position of certain
subalgebras of M, which come up due to the way M = M(D) is
constructed from the building data D.
11/13
About deformation-rigidity arguments
Each one of these results is proved by using deformation rigidity
techniques. This approach exploits the tension between “soft” and “rigid”
parts of the ambient II1 factor M, to locate the position of certain
subalgebras of M, which come up due to the way M = M(D) is
constructed from the building data D.
The softness is “measured/gaged” by deformations of M by completely
positive, or more generally completely bounded maps Φn : M → M (i.e.,
satisfying limn kΦn (x) − xk2 = 0, ∀x ∈ M). The rigidity often comes from
properties of a group involved in the construction (e.g., various forms of
property (T) of Kazhdan), or by the way certain subalgebras B ⊂ M “sit”
inside M (e.g., with spectral gap).
11/13
A double role for “structure-versus-randomness”
If such combination is “tight enough”, then the rigid part Q ⊂ M is forced
to lie in a certain precise position P inside M. This translates into the fact
that there is an intertwining bimodule Q HP ⊂ L2 (M) with
dim(HP ) < ∞, a situation one denotes by Q ≺M P.
12/13
A double role for “structure-versus-randomness”
If such combination is “tight enough”, then the rigid part Q ⊂ M is forced
to lie in a certain precise position P inside M. This translates into the fact
that there is an intertwining bimodule Q HP ⊂ L2 (M) with
dim(HP ) < ∞, a situation one denotes by Q ≺M P.
The subordination Q ≺M P means that AdU(Q) y M has a non-zero part
that’s compact relative to P. So the opposite case, Q 6≺M P, means this
action is weak mixing relative to P: ∃un ∈ U(Q) with
limn kEP (xun y )k2 = 0, ∀x, y ∈ M. So Q ≺M P, Q 6≺M P is a “structure
versus randomness” type paradigm. This is very useful in proofs “by
contradiction”, when combined with deformation-rigidity arguments.
12/13
A double role for “structure-versus-randomness”
If such combination is “tight enough”, then the rigid part Q ⊂ M is forced
to lie in a certain precise position P inside M. This translates into the fact
that there is an intertwining bimodule Q HP ⊂ L2 (M) with
dim(HP ) < ∞, a situation one denotes by Q ≺M P.
The subordination Q ≺M P means that AdU(Q) y M has a non-zero part
that’s compact relative to P. So the opposite case, Q 6≺M P, means this
action is weak mixing relative to P: ∃un ∈ U(Q) with
limn kEP (xun y )k2 = 0, ∀x, y ∈ M. So Q ≺M P, Q 6≺M P is a “structure
versus randomness” type paradigm. This is very useful in proofs “by
contradiction”, when combined with deformation-rigidity arguments.
The “rigid/soft” dichotomy fits into the structure/randomness pattern as
well, with the rigid part corresponding to “structure”, and the
soft-deformable part to “random”.
12/13
About the free group factor problem
• We mentioned Bowen’s 2009 amplification formula for the group factors
L(Γn ), with Γn = Z o Fn : L(Γn )1/k ' L(Γm ), where m = k(n − 1) + 1.
13/13
About the free group factor problem
• We mentioned Bowen’s 2009 amplification formula for the group factors
L(Γn ), with Γn = Z o Fn : L(Γn )1/k ' L(Γm ), where m = k(n − 1) + 1.
• For the free group factors L(Fn ) one has Voiculescu’s amplification
formula (1989): L(Fn )1/k ' L(Fm ), where m = k 2 (n − 1) + 1 !
Proof uses a random matrix model for L(Fn ) (free probability) !
13/13
About the free group factor problem
• We mentioned Bowen’s 2009 amplification formula for the group factors
L(Γn ), with Γn = Z o Fn : L(Γn )1/k ' L(Γm ), where m = k(n − 1) + 1.
• For the free group factors L(Fn ) one has Voiculescu’s amplification
formula (1989): L(Fn )1/k ' L(Fm ), where m = k 2 (n − 1) + 1 !
Proof uses a random matrix model for L(Fn ) (free probability) !
Extended by Radulescu, Dykema (1993): for all ∞ ≥ n, m ≥ 2 and all
t > 0 one has L(Fn )t ' L(Fm )s , where (n − 1)/t 2 = (m − 1)/s 2 . Also,
L(Fn ) 6' L(Fm ) for some 2 ≤ n < m ≤ ∞, iff L(Fn ) are all non-isomorphic,
2 ≤ n ≤ ∞.
13/13
About the free group factor problem
• We mentioned Bowen’s 2009 amplification formula for the group factors
L(Γn ), with Γn = Z o Fn : L(Γn )1/k ' L(Γm ), where m = k(n − 1) + 1.
• For the free group factors L(Fn ) one has Voiculescu’s amplification
formula (1989): L(Fn )1/k ' L(Fm ), where m = k 2 (n − 1) + 1 !
Proof uses a random matrix model for L(Fn ) (free probability) !
Extended by Radulescu, Dykema (1993): for all ∞ ≥ n, m ≥ 2 and all
t > 0 one has L(Fn )t ' L(Fm )s , where (n − 1)/t 2 = (m − 1)/s 2 . Also,
L(Fn ) 6' L(Fm ) for some 2 ≤ n < m ≤ ∞, iff L(Fn ) are all non-isomorphic,
2 ≤ n ≤ ∞.
Key open problems
• L(Fn ) ' L(Fm ) implies n = m ?
• Is L(F∞ ) infinitely generated ? (if so, then all L(Fn ) follow non-iso)
Obs There are no known examples of infinitely generated II1 factors !!
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