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Free Entropy Dimension (Voiculescu)G = ⟨S⟩ �nitely generated discrete group, τG : CF|S| → C.∏ω
Mn×n =∏∞
k=1Mk×k/{(xk) : limk→ω1
kTr(x∗k xk) = 0}
Γ(G ) = τG -preserving homomorhisms G →ω∏
Mn×n
We'll always assume that Γ(G ) ̸= ∅
Examples
(i)G
residually
�nite,
πk
:G
→G
/Gk→
Mnk×nk
=⇒
π:g7→
πk(g
)∈M
nk×nk
,π∈
Γ(G
)(i')g7→
uπ(g
)u∗ ,
u∈
∏ ω Mn×n
(ii)G
=F m
=⇒
π:(g
1,.
..,g
m)7→
(u1,.
..,u
m)arandom
m-tuplein
U(n
),
π∈
Γ(G
)(iii)
πi:Gi→
∏ ω Mn×n
=⇒
π(g
1)
=π1(g
1),
π(g
2)
=uπ2(g
2)u
∗ ,u∈U
(n)
random
,g i
∈Gi,
π∈
Γ(G
1∗G2).
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
2/16
Free Entropy Dimension (Voiculescu)G = ⟨S⟩ �nitely generated discrete group, τG : CF|S| → C.∏ω
Mn×n =∏∞
k=1Mk×k/{(xk) : limk→ω1
kTr(x∗k xk) = 0}
Γ(G ) = τG -preserving homomorhisms G →ω∏
Mn×n
We'll always assume that Γ(G ) ̸= ∅
Examples
(i)G
residually
�nite,
πk
:G
→G
/Gk→
Mnk×nk
=⇒
π:g7→
πk(g
)∈M
nk×nk
,π∈
Γ(G
)(i')g7→
uπ(g
)u∗ ,
u∈
∏ ω Mn×n
(ii)G
=F m
=⇒
π:(g
1,.
..,g
m)7→
(u1,.
..,u
m)arandom
m-tuplein
U(n
),
π∈
Γ(G
)(iii)
πi:Gi→
∏ ω Mn×n
=⇒
π(g
1)
=π1(g
1),
π(g
2)
=uπ2(g
2)u
∗ ,u∈U
(n)
random
,g i
∈Gi,
π∈
Γ(G
1∗G2).
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
2/16
Free Entropy Dimension (Voiculescu)G = ⟨S⟩ �nitely generated discrete group, τG : CF|S| → C.∏ω
Mn×n =∏∞
k=1Mk×k/{(xk) : limk→ω1
kTr(x∗k xk) = 0}
Γ(G ) = τG -preserving homomorhisms G →ω∏
Mn×n
We'll always assume that Γ(G ) ̸= ∅
Examples
(i)G
residually
�nite,
πk
:G
→G
/Gk→
Mnk×nk
=⇒
π:g7→
πk(g
)∈M
nk×nk
,π∈
Γ(G
)(i')g7→
uπ(g
)u∗ ,
u∈
∏ ω Mn×n
(ii)G
=F m
=⇒
π:(g
1,.
..,g
m)7→
(u1,.
..,u
m)arandom
m-tuplein
U(n
),
π∈
Γ(G
)(iii)
πi:Gi→
∏ ω Mn×n
=⇒
π(g
1)
=π1(g
1),
π(g
2)
=uπ2(g
2)u
∗ ,u∈U
(n)
random
,g i
∈Gi,
π∈
Γ(G
1∗G2).
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
2/16
Free Entropy Dimension (Voiculescu)G = ⟨S⟩ �nitely generated discrete group, τG : CF|S| → C.∏ω
Mn×n =∏∞
k=1Mk×k/{(xk) : limk→ω1
kTr(x∗k xk) = 0}
Γ(G ) = τG -preserving homomorhisms G →ω∏
Mn×n
We'll always assume that Γ(G ) ̸= ∅
Examples
(i)G
residually
�nite,
πk
:G
→G
/Gk→
Mnk×nk
=⇒
π:g7→
πk(g
)∈M
nk×nk
,π∈
Γ(G
)(i')g7→
uπ(g
)u∗ ,
u∈
∏ ω Mn×n
(ii)G
=F m
=⇒
π:(g
1,.
..,g
m)7→
(u1,.
..,u
m)arandom
m-tuplein
U(n
),
π∈
Γ(G
)(iii)
πi:Gi→
∏ ω Mn×n
=⇒
π(g
1)
=π1(g
1),
π(g
2)
=uπ2(g
2)u
∗ ,u∈U
(n)
random
,g i
∈Gi,
π∈
Γ(G
1∗G2).
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
2/16
Free Entropy Dimension (Voiculescu)G = ⟨S⟩ �nitely generated discrete group, τG : CF|S| → C.∏ω
Mn×n =∏∞
k=1Mk×k/{(xk) : limk→ω1
kTr(x∗k xk) = 0}
Γ(G ) = τG -preserving homomorhisms G →ω∏
Mn×n
We'll always assume that Γ(G ) ̸= ∅
Examples
(i)G
residually
�nite,
πk
:G
→G
/Gk→
Mnk×nk
=⇒
π:g7→
πk(g
)∈M
nk×nk
,π∈
Γ(G
)(i')g7→
uπ(g
)u∗ ,
u∈
∏ ω Mn×n
(ii)G
=F m
=⇒
π:(g
1,.
..,g
m)7→
(u1,.
..,u
m)arandom
m-tuplein
U(n
),
π∈
Γ(G
)(iii)
πi:Gi→
∏ ω Mn×n
=⇒
π(g
1)
=π1(g
1),
π(g
2)
=uπ2(g
2)u
∗ ,u∈U
(n)
random
,g i
∈Gi,
π∈
Γ(G
1∗G2).
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
2/16
Free Entropy Dimension (Voiculescu)G = ⟨S⟩ �nitely generated discrete group, τG : CF|S| → C.∏ω
Mn×n =∏∞
k=1Mk×k/{(xk) : limk→ω1
kTr(x∗k xk) = 0}
Γ(G ) = τG -preserving homomorhisms G →ω∏
Mn×n
We'll always assume that Γ(G ) ̸= ∅
Examples
(i)G
residually
�nite,
πk
:G
→G
/Gk→
Mnk×nk
=⇒
π:g7→
πk(g
)∈M
nk×nk
,π∈
Γ(G
)(i')g7→
uπ(g
)u∗ ,
u∈
∏ ω Mn×n
(ii)G
=F m
=⇒
π:(g
1,.
..,g
m)7→
(u1,.
..,u
m)arandom
m-tuplein
U(n
),
π∈
Γ(G
)(iii)
πi:Gi→
∏ ω Mn×n
=⇒
π(g
1)
=π1(g
1),
π(g
2)
=uπ2(g
2)u
∗ ,u∈U
(n)
random
,g i
∈Gi,
π∈
Γ(G
1∗G2).
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
2/16
De�nition of δ(G )G = ⟨S⟩ �nitely generated discrete group, τG : CF|S| → C.∏ω
Mn×n =∏∞
k=1Mk×k/{(xk) : limk→ω1
kTr(x∗k xk) = 0}
Γ(G ) = τG -preserving homomorhisms G →ω∏
Mn×n
δ(G ) = normalized covering dimension of Γ(G )
De�nition
S⊂
Gsetofgenerators.
Γ(S
;n,l
,ϵ)
={(ug) g
∈S⊂
U(n
)|S|:1 nTr(p((ug) g
∈S))
≈ϵτ G
(p)}
forallp∈
F |S|of
length≤
l.
δ(G
)=
δ(S)
=lim
sup
r→0
inf
ϵ,llim
sup
n→
∞
logr-coveringnumber
ofΓ(S
;n,l
,ϵ)
n2logr
NB:δ(S)makes
sense
foranySin
atracialsubalgebraof
∏ ω Mn×n
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
3/16
De�nition of δ(G )G = ⟨S⟩ �nitely generated discrete group, τG : CF|S| → C.∏ω
Mn×n =∏∞
k=1Mk×k/{(xk) : limk→ω1
kTr(x∗k xk) = 0}
Γ(G ) = τG -preserving homomorhisms G →ω∏
Mn×n
δ(G ) = normalized covering dimension of Γ(G )
De�nition
S⊂
Gsetofgenerators.
Γ(S
;n,l
,ϵ)
={(ug) g
∈S⊂
U(n
)|S|:1 nTr(p((ug) g
∈S))
≈ϵτ G
(p)}
forallp∈
F |S|of
length≤
l.
δ(G
)=
δ(S)
=lim
sup
r→0
inf
ϵ,llim
sup
n→
∞
logr-coveringnumber
ofΓ(S
;n,l
,ϵ)
n2logr
NB:δ(S)makes
sense
foranySin
atracialsubalgebraof
∏ ω Mn×n
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
3/16
De�nition of δ(G )G = ⟨S⟩ �nitely generated discrete group, τG : CF|S| → C.∏ω
Mn×n =∏∞
k=1Mk×k/{(xk) : limk→ω1
kTr(x∗k xk) = 0}
Γ(G ) = τG -preserving homomorhisms G →ω∏
Mn×n
δ(G ) = normalized covering dimension of Γ(G )
De�nition
S⊂
Gsetofgenerators.
Γ(S
;n,l
,ϵ)
={(ug) g
∈S⊂
U(n
)|S|:1 nTr(p((ug) g
∈S))
≈ϵτ G
(p)}
forallp∈
F |S|of
length≤
l.
δ(G
)=
δ(S)
=lim
sup
r→0
inf
ϵ,llim
sup
n→
∞
logr-coveringnumber
ofΓ(S
;n,l
,ϵ)
n2logr
NB:δ(S)makes
sense
foranySin
atracialsubalgebraof
∏ ω Mn×n
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
3/16
De�nition of δ(G )G = ⟨S⟩ �nitely generated discrete group, τG : CF|S| → C.∏ω
Mn×n =∏∞
k=1Mk×k/{(xk) : limk→ω1
kTr(x∗k xk) = 0}
Γ(G ) = τG -preserving homomorhisms G →ω∏
Mn×n
δ(G ) = normalized covering dimension of Γ(G )
De�nition
S⊂
Gsetofgenerators.
Γ(S
;n,l
,ϵ)
={(ug) g
∈S⊂
U(n
)|S|:1 nTr(p((ug) g
∈S))
≈ϵτ G
(p)}
forallp∈
F |S|of
length≤
l.
δ(G
)=
δ(S)
=lim
sup
r→0
inf
ϵ,llim
sup
n→
∞
logr-coveringnumber
ofΓ(S
;n,l
,ϵ)
n2logr
NB:δ(S)makes
sense
foranySin
atracialsubalgebraof
∏ ω Mn×n
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
3/16
Properties of δ(G )
You can think of Γ(G ) as an analog of the setA(G ) = {homomorphisms G → Aut(X , µ)}.
Example
G=
Z(oranyin�niteam
enablegroup).
Then
π1,π
2∈
Γ(G
)=⇒
∃u∈
∏ ω Mn×nso
that
π1(g
)=
u1π2(g
)u∗ 2.
δisnormalized
sothat
δ(G
)=
1in
thiscase.
Theorem
[K.Jung]Γ(G)/(conjugationbyunitariesin∏ωMn×n)={pt}characterizes
amenability.
DimaShlyakhtenko(UCLA)SomeappicationsoffreestochasticcalculustoII1factors.4/16
Properties of δ(G )
You can think of Γ(G ) as an analog of the setA(G ) = {homomorphisms G → Aut(X , µ)}.
Example
G=
Z(oranyin�niteam
enablegroup).
Then
π1,π
2∈
Γ(G
)=⇒
∃u∈
∏ ω Mn×nso
that
π1(g
)=
u1π2(g
)u∗ 2.
δisnormalized
sothat
δ(G
)=
1in
thiscase.
Theorem
[K.Jung]Γ(G)/(conjugationbyunitariesin∏ωMn×n)={pt}characterizes
amenability.
DimaShlyakhtenko(UCLA)SomeappicationsoffreestochasticcalculustoII1factors.4/16
Properties of δ(G )
You can think of Γ(G ) as an analog of the setA(G ) = {homomorphisms G → Aut(X , µ)}.
Example
G=
Z(oranyin�niteam
enablegroup).
Then
π1,π
2∈
Γ(G
)=⇒
∃u∈
∏ ω Mn×nso
that
π1(g
)=
u1π2(g
)u∗ 2.
δisnormalized
sothat
δ(G
)=
1in
thiscase.
Theorem
[K.Jung]Γ(G)/(conjugationbyunitariesin∏ωMn×n)={pt}characterizes
amenability.
DimaShlyakhtenko(UCLA)SomeappicationsoffreestochasticcalculustoII1factors.4/16
More on δ(G )
Theorem
[K.Jung+
D.S.]Sgeneratingsetof
aproperty
(T)II1factor
M=⇒
δ(S)
=1.
Inparticular,if
Γhas
property
(T)
=⇒
δ(Γ)
=1.
Proof.
Γ(S)/(conjugation)isdiscrete.
Question.Istheconversetrue?DoesGhave(T)i�Γ(G)/(conjugation)isdiscrete?Relatedquestion:replaceΓ(G)withthesetofτGpreservinghomomorphismsintoallpossibleII1factors.DoesGhave(T)i�thisset(uptounitaryconjugacy)isdiscrete?
DimaShlyakhtenko(UCLA)SomeappicationsoffreestochasticcalculustoII1factors.5/16
More on δ(G )
Theorem
[K.Jung+
D.S.]Sgeneratingsetof
aproperty
(T)II1factor
M=⇒
δ(S)
=1.
Inparticular,if
Γhas
property
(T)
=⇒
δ(Γ)
=1.
Proof.
Γ(S)/(conjugation)isdiscrete.
Question.Istheconversetrue?DoesGhave(T)i�Γ(G)/(conjugation)isdiscrete?Relatedquestion:replaceΓ(G)withthesetofτGpreservinghomomorphismsintoallpossibleII1factors.DoesGhave(T)i�thisset(uptounitaryconjugacy)isdiscrete?
DimaShlyakhtenko(UCLA)SomeappicationsoffreestochasticcalculustoII1factors.5/16
More on δ(G )
Theorem
[K.Jung+
D.S.]Sgeneratingsetof
aproperty
(T)II1factor
M=⇒
δ(S)
=1.
Inparticular,if
Γhas
property
(T)
=⇒
δ(Γ)
=1.
Proof.
Γ(S)/(conjugation)isdiscrete.
Question.Istheconversetrue?DoesGhave(T)i�Γ(G)/(conjugation)isdiscrete?Relatedquestion:replaceΓ(G)withthesetofτGpreservinghomomorphismsintoallpossibleII1factors.DoesGhave(T)i�thisset(uptounitaryconjugacy)isdiscrete?
DimaShlyakhtenko(UCLA)SomeappicationsoffreestochasticcalculustoII1factors.5/16
More on δ(G )
Theorem
[K.Jung+
D.S.]Sgeneratingsetof
aproperty
(T)II1factor
M=⇒
δ(S)
=1.
Inparticular,if
Γhas
property
(T)
=⇒
δ(Γ)
=1.
Proof.
Γ(S)/(conjugation)isdiscrete.
Question.Istheconversetrue?DoesGhave(T)i�Γ(G)/(conjugation)isdiscrete?Relatedquestion:replaceΓ(G)withthesetofτGpreservinghomomorphismsintoallpossibleII1factors.DoesGhave(T)i�thisset(uptounitaryconjugacy)isdiscrete?
DimaShlyakhtenko(UCLA)SomeappicationsoffreestochasticcalculustoII1factors.5/16
Vanishing results.
There are by now many theorems that imply that under some conditions, δ mustbe 1.
In other words, Γ(G )/(conjugation by unitaries in G ) is �small�.
Theorem [Voiuculescu]W
∗ (S)has
aCartansubalgebra
=⇒
δ(S)
=1
[Ge]
W∗ (S)
=M1⊗M2withM
jdi�use
=⇒
δ(S)
=1
[Stefan,Ge-Shen,Jung]W
∗ (S)
=spanR1A1R2..
.AkRk,Aj�nitesubsets,Rk
hyper�nite
=⇒
δ(S)sm
all.
Conjecture.
δ(G
)=
β(2
)1
(G)−
β(2
)0
(G)+1
β(2
)1
(G)−
β(2
)0
(G)+1
=dim
L(G
){spaceof
ℓ2(G
)-valuedcocycles}
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
6/16
Vanishing results.
There are by now many theorems that imply that under some conditions, δ mustbe 1.
In other words, Γ(G )/(conjugation by unitaries in G ) is �small�.
Theorem [Voiuculescu]W
∗ (S)has
aCartansubalgebra
=⇒
δ(S)
=1
[Ge]
W∗ (S)
=M1⊗M2withM
jdi�use
=⇒
δ(S)
=1
[Stefan,Ge-Shen,Jung]W
∗ (S)
=spanR1A1R2..
.AkRk,Aj�nitesubsets,Rk
hyper�nite
=⇒
δ(S)sm
all.
Conjecture.
δ(G
)=
β(2
)1
(G)−
β(2
)0
(G)+1
β(2
)1
(G)−
β(2
)0
(G)+1
=dim
L(G
){spaceof
ℓ2(G
)-valuedcocycles}
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
6/16
Vanishing results.
There are by now many theorems that imply that under some conditions, δ mustbe 1.
In other words, Γ(G )/(conjugation by unitaries in G ) is �small�.
Theorem [Voiuculescu]W
∗ (S)has
aCartansubalgebra
=⇒
δ(S)
=1
[Ge]
W∗ (S)
=M1⊗M2withM
jdi�use
=⇒
δ(S)
=1
[Stefan,Ge-Shen,Jung]W
∗ (S)
=spanR1A1R2..
.AkRk,Aj�nitesubsets,Rk
hyper�nite
=⇒
δ(S)sm
all.
Conjecture.
δ(G
)=
β(2
)1
(G)−
β(2
)0
(G)+1
β(2
)1
(G)−
β(2
)0
(G)+1
=dim
L(G
){spaceof
ℓ2(G
)-valuedcocycles}
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
6/16
Vanishing results.
There are by now many theorems that imply that under some conditions, δ mustbe 1.
In other words, Γ(G )/(conjugation by unitaries in G ) is �small�.
Theorem [Voiuculescu]W
∗ (S)has
aCartansubalgebra
=⇒
δ(S)
=1
[Ge]
W∗ (S)
=M1⊗M2withM
jdi�use
=⇒
δ(S)
=1
[Stefan,Ge-Shen,Jung]W
∗ (S)
=spanR1A1R2..
.AkRk,Aj�nitesubsets,Rk
hyper�nite
=⇒
δ(S)sm
all.
Conjecture.
δ(G
)=
β(2
)1
(G)−
β(2
)0
(G)+1
β(2
)1
(G)−
β(2
)0
(G)+1
=dim
L(G
){spaceof
ℓ2(G
)-valuedcocycles}
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
6/16
δ(G ) = β(2)1 (G ) − β
(2)0 (G ) + 1.
To prove the conjecture one needs:1 δ(G ) ≤ dimL(G){space of ℓ2(G )-valued cocycles} (i.e., if one has manynon-cojugate �actions� in Γ(G ), then there are non-trivial cocycles)
2 δ(G ) ≥ dimL(G){space of ℓ2(G )-valued cocycles} (i.e., if one has non-trivialcocycles, to produce many non-conjugate �actions� in Γ(G )).
Theorem
[A.Connes
+D.S.]
δ(G
)≤
dim
L(G
){spaceof
ℓ2(G
)-valued
cocycles}
Lookshardat
�rst:
how
does
oneproduce
cocycles
from
know
ingtherearemany
elem
ents
inΓ(S
)?Idea:dim
L(G
){sp
ace
ofℓ2
(G)-va
lued
cocy
cles}sm
all⇔
thereare
manycy
cles
in(ord
inary
!)homology.
Veryvaguely:
thereason
β(2
)1
(G)issm
allisthat
therearemanyrelationsam
ong
generatorsof
G=⇒
restrictionsonhow
manyelem
ents
theremay
bein
Γ(G
).Realproof
uses:
largedeviationsupper
boundsforrandom
matrices(B
iane,
Capitaine,
Guionnet)+
non-m
icrostates
free
entropydim
ension+
.....
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
7/16
δ(G ) = β(2)1 (G ) − β
(2)0 (G ) + 1.
To prove the conjecture one needs:1 δ(G ) ≤ dimL(G){space of ℓ2(G )-valued cocycles} (i.e., if one has manynon-cojugate �actions� in Γ(G ), then there are non-trivial cocycles)
2 δ(G ) ≥ dimL(G){space of ℓ2(G )-valued cocycles} (i.e., if one has non-trivialcocycles, to produce many non-conjugate �actions� in Γ(G )).
Theorem
[A.Connes
+D.S.]
δ(G
)≤
dim
L(G
){spaceof
ℓ2(G
)-valued
cocycles}
Lookshardat
�rst:
how
does
oneproduce
cocycles
from
know
ingtherearemany
elem
ents
inΓ(S
)?Idea:dim
L(G
){sp
ace
ofℓ2
(G)-va
lued
cocy
cles}sm
all⇔
thereare
manycy
cles
in(ord
inary
!)homology.
Veryvaguely:
thereason
β(2
)1
(G)issm
allisthat
therearemanyrelationsam
ong
generatorsof
G=⇒
restrictionsonhow
manyelem
ents
theremay
bein
Γ(G
).Realproof
uses:
largedeviationsupper
boundsforrandom
matrices(B
iane,
Capitaine,
Guionnet)+
non-m
icrostates
free
entropydim
ension+
.....
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
7/16
δ(G ) = β(2)1 (G ) − β
(2)0 (G ) + 1.
To prove the conjecture one needs:1 δ(G ) ≤ dimL(G){space of ℓ2(G )-valued cocycles} (i.e., if one has manynon-cojugate �actions� in Γ(G ), then there are non-trivial cocycles)
2 δ(G ) ≥ dimL(G){space of ℓ2(G )-valued cocycles} (i.e., if one has non-trivialcocycles, to produce many non-conjugate �actions� in Γ(G )).
Theorem
[A.Connes
+D.S.]
δ(G
)≤
dim
L(G
){spaceof
ℓ2(G
)-valued
cocycles}
Lookshardat
�rst:
how
does
oneproduce
cocycles
from
know
ingtherearemany
elem
ents
inΓ(S
)?Idea:dim
L(G
){sp
ace
ofℓ2
(G)-va
lued
cocy
cles}sm
all⇔
thereare
manycy
cles
in(ord
inary
!)homology.
Veryvaguely:
thereason
β(2
)1
(G)issm
allisthat
therearemanyrelationsam
ong
generatorsof
G=⇒
restrictionsonhow
manyelem
ents
theremay
bein
Γ(G
).Realproof
uses:
largedeviationsupper
boundsforrandom
matrices(B
iane,
Capitaine,
Guionnet)+
non-m
icrostates
free
entropydim
ension+
.....
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
7/16
δ(G ) = β(2)1 (G ) − β
(2)0 (G ) + 1.
To prove the conjecture one needs:1 δ(G ) ≤ dimL(G){space of ℓ2(G )-valued cocycles} (i.e., if one has manynon-cojugate �actions� in Γ(G ), then there are non-trivial cocycles)
2 δ(G ) ≥ dimL(G){space of ℓ2(G )-valued cocycles} (i.e., if one has non-trivialcocycles, to produce many non-conjugate �actions� in Γ(G )).
Theorem
[A.Connes
+D.S.]
δ(G
)≤
dim
L(G
){spaceof
ℓ2(G
)-valued
cocycles}
Lookshardat
�rst:
how
does
oneproduce
cocycles
from
know
ingtherearemany
elem
ents
inΓ(S
)?Idea:dim
L(G
){sp
ace
ofℓ2
(G)-va
lued
cocy
cles}sm
all⇔
thereare
manycy
cles
in(ord
inary
!)homology.
Veryvaguely:
thereason
β(2
)1
(G)issm
allisthat
therearemanyrelationsam
ong
generatorsof
G=⇒
restrictionsonhow
manyelem
ents
theremay
bein
Γ(G
).Realproof
uses:
largedeviationsupper
boundsforrandom
matrices(B
iane,
Capitaine,
Guionnet)+
non-m
icrostates
free
entropydim
ension+
.....
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
7/16
δ(G ) = β(2)1 (G ) − β
(2)0 (G ) + 1.
To prove the conjecture one needs:1 δ(G ) ≤ dimL(G){space of ℓ2(G )-valued cocycles} (i.e., if one has manynon-cojugate �actions� in Γ(G ), then there are non-trivial cocycles)
2 δ(G ) ≥ dimL(G){space of ℓ2(G )-valued cocycles} (i.e., if one has non-trivialcocycles, to produce many non-conjugate �actions� in Γ(G )).
Theorem
[A.Connes
+D.S.]
δ(G
)≤
dim
L(G
){spaceof
ℓ2(G
)-valued
cocycles}
Lookshardat
�rst:
how
does
oneproduce
cocycles
from
know
ingtherearemany
elem
ents
inΓ(S
)?Idea:dim
L(G
){sp
ace
ofℓ2
(G)-va
lued
cocy
cles}sm
all⇔
thereare
manycy
cles
in(ord
inary
!)homology.
Veryvaguely:
thereason
β(2
)1
(G)issm
allisthat
therearemanyrelationsam
ong
generatorsof
G=⇒
restrictionsonhow
manyelem
ents
theremay
bein
Γ(G
).Realproof
uses:
largedeviationsupper
boundsforrandom
matrices(B
iane,
Capitaine,
Guionnet)+
non-m
icrostates
free
entropydim
ension+
.....
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
7/16
δ(G ) = β(2)1 (G ) − β
(2)0 (G ) + 1.
To prove the conjecture one needs:1 δ(G ) ≤ dimL(G){space of ℓ2(G )-valued cocycles} (i.e., if one has manynon-cojugate �actions� in Γ(G ), then there are non-trivial cocycles)
2 δ(G ) ≥ dimL(G){space of ℓ2(G )-valued cocycles} (i.e., if one has non-trivialcocycles, to produce many non-conjugate �actions� in Γ(G )).
Theorem
[A.Connes
+D.S.]
δ(G
)≤
dim
L(G
){spaceof
ℓ2(G
)-valued
cocycles}
Lookshardat
�rst:
how
does
oneproduce
cocycles
from
know
ingtherearemany
elem
ents
inΓ(S
)?Idea:dim
L(G
){sp
ace
ofℓ2
(G)-va
lued
cocy
cles}sm
all⇔
thereare
manycy
cles
in(ord
inary
!)homology.
Veryvaguely:
thereason
β(2
)1
(G)issm
allisthat
therearemanyrelationsam
ong
generatorsof
G=⇒
restrictionsonhow
manyelem
ents
theremay
bein
Γ(G
).Realproof
uses:
largedeviationsupper
boundsforrandom
matrices(B
iane,
Capitaine,
Guionnet)+
non-m
icrostates
free
entropydim
ension+
.....
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
7/16
δ(G ) = β(2)1 (G ) − β
(2)0 (G ) + 1.
On the other hand, the second direction:
δ(G ) ≥ dimL(G){space of ℓ2(G )-valued cocycles}
looks constructive/easy.Given c : G → ℓ2(G ) construct elements in Γ(G ).
c : G → ℓ2(G ) gives rise to ∂ : CG → ℓ2(G )⊗̄ℓ2(G ).Idea: use free stochastic calculus to �exponentiate� ∂ to a deformation in thesense of Popa:Let M = L(G ) and let M = M ∗ L(F∞). Then
ML2(M)M ∼= L2(M) ⊕[L2(M) ⊗ L2(M)
]⊕∞
Given ∂ : CG → [L2(M)⊗̄L2(M)]⊕m, hope to �nd αt : M → M ∗ L(F∞) so that
αt(X ) = X + ∂(X )t + O(t2)
Dima Shlyakhtenko (UCLA) Some appications of free stochastic calculus to II1 factors. 8 / 16
δ(G ) = β(2)1 (G ) − β
(2)0 (G ) + 1.
On the other hand, the second direction:
δ(G ) ≥ dimL(G){space of ℓ2(G )-valued cocycles}
looks constructive/easy.
Given c : G → ℓ2(G ) construct elements in Γ(G ).c : G → ℓ2(G ) gives rise to ∂ : CG → ℓ2(G )⊗̄ℓ2(G ).Idea: use free stochastic calculus to �exponentiate� ∂ to a deformation in thesense of Popa:Let M = L(G ) and let M = M ∗ L(F∞). Then
ML2(M)M ∼= L2(M) ⊕[L2(M) ⊗ L2(M)
]⊕∞
Given ∂ : CG → [L2(M)⊗̄L2(M)]⊕m, hope to �nd αt : M → M ∗ L(F∞) so that
αt(X ) = X + ∂(X )t + O(t2)
Dima Shlyakhtenko (UCLA) Some appications of free stochastic calculus to II1 factors. 8 / 16
δ(G ) = β(2)1 (G ) − β
(2)0 (G ) + 1.
On the other hand, the second direction:
δ(G ) ≥ dimL(G){space of ℓ2(G )-valued cocycles}
looks constructive/easy.
Given c : G → ℓ2(G ) construct elements in Γ(G ).c : G → ℓ2(G ) gives rise to ∂ : CG → ℓ2(G )⊗̄ℓ2(G ).Idea: use free stochastic calculus to �exponentiate� ∂ to a deformation in thesense of Popa:Let M = L(G ) and let M = M ∗ L(F∞). Then
ML2(M)M ∼= L2(M) ⊕[L2(M) ⊗ L2(M)
]⊕∞
Given ∂ : CG → [L2(M)⊗̄L2(M)]⊕m, hope to �nd αt : M → M ∗ L(F∞) so that
αt(X ) = X + ∂(X )t + O(t2)
Dima Shlyakhtenko (UCLA) Some appications of free stochastic calculus to II1 factors. 8 / 16
δ(G ) = β(2)1 (G ) − β
(2)0 (G ) + 1.
On the other hand, the second direction:
δ(G ) ≥ dimL(G){space of ℓ2(G )-valued cocycles}
looks constructive/easy.
Given c : G → ℓ2(G ) construct elements in Γ(G ).c : G → ℓ2(G ) gives rise to ∂ : CG → ℓ2(G )⊗̄ℓ2(G ).Idea: use free stochastic calculus to �exponentiate� ∂ to a deformation in thesense of Popa:Let M = L(G ) and let M = M ∗ L(F∞). Then
ML2(M)M ∼= L2(M) ⊕[L2(M) ⊗ L2(M)
]⊕∞
Given ∂ : CG → [L2(M)⊗̄L2(M)]⊕m, hope to �nd αt : M → M ∗ L(F∞) so that
αt(X ) = X + ∂(X )t + O(t2)
Dima Shlyakhtenko (UCLA) Some appications of free stochastic calculus to II1 factors. 8 / 16
Deformations and estimate on δ.
Let S1, . . . ,Sm be a free semicircular family in L(F∞).
Theorem
Assumethat
X⃗=
(X1,.
..,X
n)∈
(M,τ
)and∃α
t:M
→M
∗L(F
∞)so
that
α0(x
)=
x∗1and
αt(X
j)=
Xj+
Qjt ︸︷︷︸ ∂
(Xj)
+O
(t2)
forsomeQj∈span(M
S1M
+···+
MSmM
).
Then
δ(X⃗
)≥
dim
M⊗̄M
ospan(M
Q1M
+···+
MQmM
).
Keypointoftheproof:
Given
π∈
Γ(X⃗
),π◦
αt∈
Γ(X⃗
).Ontheother
hand,
π◦
αt(X
j)−Xj=
Qj+O
(t2)andtherearemany(random
)em
beddingsofQj'is
into
∏ ω Mn.
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
9/16
Deformations and estimate on δ.
Let S1, . . . ,Sm be a free semicircular family in L(F∞).
Theorem
Assumethat
X⃗=
(X1,.
..,X
n)∈
(M,τ
)and∃α
t:M
→M
∗L(F
∞)so
that
α0(x
)=
x∗1and
αt(X
j)=
Xj+
Qjt ︸︷︷︸ ∂
(Xj)
+O
(t2)
forsomeQj∈span(M
S1M
+···+
MSmM
).
Then
δ(X⃗
)≥
dim
M⊗̄M
ospan(M
Q1M
+···+
MQmM
).
Keypointoftheproof:
Given
π∈
Γ(X⃗
),π◦
αt∈
Γ(X⃗
).Ontheother
hand,
π◦
αt(X
j)−Xj=
Qj+O
(t2)andtherearemany(random
)em
beddingsofQj'is
into
∏ ω Mn.
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
9/16
Deformations and estimate on δ.
Let S1, . . . ,Sm be a free semicircular family in L(F∞).
Theorem
Assumethat
X⃗=
(X1,.
..,X
n)∈
(M,τ
)and∃α
t:M
→M
∗L(F
∞)so
that
α0(x
)=
x∗1and
αt(X
j)=
Xj+
Qjt ︸︷︷︸ ∂
(Xj)
+O
(t2)
forsomeQj∈span(M
S1M
+···+
MSmM
).
Then
δ(X⃗
)≥
dim
M⊗̄M
ospan(M
Q1M
+···+
MQmM
).
Keypointoftheproof:
Given
π∈
Γ(X⃗
),π◦
αt∈
Γ(X⃗
).Ontheother
hand,
π◦
αt(X
j)−Xj=
Qj+O
(t2)andtherearemany(random
)em
beddingsofQj'is
into
∏ ω Mn.
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
9/16
Stochastic Calculus
Main input: Brownian motion process: family of random variables B([s, t)),t > s ≥ 0 so that B([s, t)) are Gaussian, B([s, t)) + B([t, r)) = B([s, r)) ifs < t < r and B([s, t)) is independent from B([s ′, t ′)) if [s, t) ∩ [s ′, t ′] = ∅.Then one can write
B([0, t)) =
ˆ t
0
dBt .
Furthermore, for nice enough probability measures µ, there exists a process Xt
with the properties that:
Xt is stationary, i.e., Xt has distribution µ for all t
Xt satis�es the stochastic di�erential equation dXt = ϕ(Xt) · dBt − ζ(Xt)dt
(Note: dBt ∼ O(t1/2)).
Dima Shlyakhtenko (UCLA) Some appications of free stochastic calculus to II1 factors. 10 / 16
dXt = ϕ(Xt) · dBt − ζ(Xt)dt
Very roughly, this means that
Xt+ϵ = Xt + ϕ(Xt)B[t, t + ϵ]︸ ︷︷ ︸O(ε1/2)
−ζ(Xt)ϵ + O(ϵ3/2)
In particular, the mapf (X ) 7→ f (Xt2)
gives rise to an isomoprhism αt : L∞(R, µ) → W ∗(Xt) ⊂ W ∗(X ,B[s, t] : s < t)which satis�es
αt(f (X )) = f (X ) + ϕ(X )f ′(X )B([0, t2))︸ ︷︷ ︸O(t)
+O(t2)
(i.e., it �exponentiates� the derivation ∂(f ) = ϕf ′, ∂ : polynomials → L2(R, µ)).
Dima Shlyakhtenko (UCLA) Some appications of free stochastic calculus to II1 factors. 11 / 16
Free Stochastic Di�erential Equations
Main fact: L(F∞) is generated by a family of self-adjoint elements
S⃗([0, t)) = {Sk([0, t))}nk=1 associated to intervals [0, t) ⊂ R, t ≥ 0. For all t,
S⃗([0, t)) is a free semicircular family and S⃗([0, t)) is free from
S⃗([0, t ′)) − S⃗([0, t)) if t ′ > t.
S⃗([0, t)) is the free analog of Brownian motion on Rn measured at time t.
One can write
S⃗([0, t)) =
ˆ t
0
dS⃗t
One can consider the free SDE
dX⃗t = Φ(Xt)#dS⃗t − ξ(Xt)dt
Here Φ(Xt) ∈ Mn×n(W∗(Xt) ⊗W ∗(Xt)). For A ∈ Mn×n(W
∗(Xt) ⊗W ∗(Xt)),
A#dS⃗ =∑
Aij#Sj , and for A = a⊗ b ∈ W ∗(Xt)⊗W ∗(Xt), A#dSj = a · dSj · b.
Dima Shlyakhtenko (UCLA) Some appications of free stochastic calculus to II1 factors. 12 / 16
Free Stochastic Di�erential Equations
Main fact: L(F∞) is generated by a family of self-adjoint elements
S⃗([0, t)) = {Sk([0, t))}nk=1 associated to intervals [0, t) ⊂ R, t ≥ 0. For all t,
S⃗([0, t)) is a free semicircular family and S⃗([0, t)) is free from
S⃗([0, t ′)) − S⃗([0, t)) if t ′ > t.
S⃗([0, t)) is the free analog of Brownian motion on Rn measured at time t.One can write
S⃗([0, t)) =
ˆ t
0
dS⃗t
One can consider the free SDE
dX⃗t = Φ(Xt)#dS⃗t − ξ(Xt)dt
Here Φ(Xt) ∈ Mn×n(W∗(Xt) ⊗W ∗(Xt)). For A ∈ Mn×n(W
∗(Xt) ⊗W ∗(Xt)),
A#dS⃗ =∑
Aij#Sj , and for A = a⊗ b ∈ W ∗(Xt)⊗W ∗(Xt), A#dSj = a · dSj · b.
Dima Shlyakhtenko (UCLA) Some appications of free stochastic calculus to II1 factors. 12 / 16
Free Stochastic Di�erential Equations
Main fact: L(F∞) is generated by a family of self-adjoint elements
S⃗([0, t)) = {Sk([0, t))}nk=1 associated to intervals [0, t) ⊂ R, t ≥ 0. For all t,
S⃗([0, t)) is a free semicircular family and S⃗([0, t)) is free from
S⃗([0, t ′)) − S⃗([0, t)) if t ′ > t.
S⃗([0, t)) is the free analog of Brownian motion on Rn measured at time t.One can write
S⃗([0, t)) =
ˆ t
0
dS⃗t
One can consider the free SDE
dX⃗t = Φ(Xt)#dS⃗t − ξ(Xt)dt
Here Φ(Xt) ∈ Mn×n(W∗(Xt) ⊗W ∗(Xt)). For A ∈ Mn×n(W
∗(Xt) ⊗W ∗(Xt)),
A#dS⃗ =∑
Aij#Sj , and for A = a⊗ b ∈ W ∗(Xt)⊗W ∗(Xt), A#dSj = a · dSj · b.
Dima Shlyakhtenko (UCLA) Some appications of free stochastic calculus to II1 factors. 12 / 16
Stationary solutions.
If Xt is a stationary solution (this means that ∂tτ(f (X⃗t)) = 0 for allnon-commutative polynomials f ), then the map
αt2 : f (X⃗ ) 7→ f (X⃗t2)
extends to an isomorphism from W ∗(X⃗ ) to W ∗(Xt) ⊂ W ∗(X⃗ , S⃗([s, t)) : s < t).Moreover,
αt(Xj) = Xj + ∂(Xj)#dS⃗t2 + O(t2).
so we can apply the estimate on δ.
Theorem
Let
X⃗beasetof
self-adjointelem
ents.Assumethat
dX⃗t=
Φ(X⃗
t)#
dS⃗t−
ξ(Xt)dthas
astationarysolution
sothat
X⃗0
=X⃗.Then
δ(X⃗
)≥
dim
M⊗̄M
o(imageof
thematrix
Φ(X⃗
)).
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
13/16
Existence of stationary solutions.
.... is a somewhat subtle problem in the non-commutative case.
The commutative case relies on equivalence between SDE and PDE + propertiesof elliptic PDEs.Not clear what happens in the non-commutative case.Best result so far:
Theorem
Thefree
SDE
dX⃗t=
Φ(X
t)#
dS⃗t−
ξ(Xt)dt
has
astationarysolution
provided
that
Φand
ξareanalytic(given
bypow
er
series)and
ξ=
∂∗ ∂
(X⃗)where
∂(X
j)=
∑ iΦjiSi.
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
14/16
Existence of stationary solutions.
.... is a somewhat subtle problem in the non-commutative case.The commutative case relies on equivalence between SDE and PDE + propertiesof elliptic PDEs.Not clear what happens in the non-commutative case.Best result so far:
Theorem
Thefree
SDE
dX⃗t=
Φ(X
t)#
dS⃗t−
ξ(Xt)dt
has
astationarysolution
provided
that
Φand
ξareanalytic(given
bypow
er
series)and
ξ=
∂∗ ∂
(X⃗)where
∂(X
j)=
∑ iΦjiSi.
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
14/16
Existence of stationary solutions.
.... is a somewhat subtle problem in the non-commutative case.The commutative case relies on equivalence between SDE and PDE + propertiesof elliptic PDEs.Not clear what happens in the non-commutative case.Best result so far:
Theorem
Thefree
SDE
dX⃗t=
Φ(X
t)#
dS⃗t−
ξ(Xt)dt
has
astationarysolution
provided
that
Φand
ξareanalytic(given
bypow
er
series)and
ξ=
∂∗ ∂
(X⃗)where
∂(X
j)=
∑ iΦjiSi.
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
14/16
Existence of stationary solutions.
.... is a somewhat subtle problem in the non-commutative case.The commutative case relies on equivalence between SDE and PDE + propertiesof elliptic PDEs.Not clear what happens in the non-commutative case.Best result so far:
Theorem
Thefree
SDE
dX⃗t=
Φ(X
t)#
dS⃗t−
ξ(Xt)dt
has
astationarysolution
provided
that
Φand
ξareanalytic(given
bypow
er
series)and
ξ=
∂∗ ∂
(X⃗)where
∂(X
j)=
∑ iΦjiSi.
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
14/16
Group case.
As a corollary you get:
δ(G ) = β(2)1
(G ) − β(2)0
(G ) − 1
provided that H1(G ; ℓ2(G )) can be generated by a cocycle whose values at thegenerators of G are �analytic� (i.e., are convergent non-commutative power seriesrather than arbitrary elements of ℓ2).
Examples
Freegroups;groupswith
β(2
)1
=0;
free
products,�nite-index
extensions/subgroups,....
Can
probably
handlegroupswhere
P:ℓ2
(G)N
→{values
ofℓ2
cocycles
ongenerators}
∈M
N×N
CG
holom.
Thereishop
eto
improvetheanalysisneeded
forexistence
ofstationarysolutions.
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
15/16
Group case.
As a corollary you get:
δ(G ) = β(2)1
(G ) − β(2)0
(G ) − 1
provided that H1(G ; ℓ2(G )) can be generated by a cocycle whose values at thegenerators of G are �analytic� (i.e., are convergent non-commutative power seriesrather than arbitrary elements of ℓ2).
Examples
Freegroups;groupswith
β(2
)1
=0;
free
products,�nite-index
extensions/subgroups,....
Can
probably
handlegroupswhere
P:ℓ2
(G)N
→{values
ofℓ2
cocycles
ongenerators}
∈M
N×N
CG
holom.
Thereishop
eto
improvetheanalysisneeded
forexistence
ofstationarysolutions.
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
15/16
Group case.
As a corollary you get:
δ(G ) = β(2)1
(G ) − β(2)0
(G ) − 1
provided that H1(G ; ℓ2(G )) can be generated by a cocycle whose values at thegenerators of G are �analytic� (i.e., are convergent non-commutative power seriesrather than arbitrary elements of ℓ2).
Examples
Freegroups;groupswith
β(2
)1
=0;
free
products,�nite-index
extensions/subgroups,....
Can
probably
handlegroupswhere
P:ℓ2
(G)N
→{values
ofℓ2
cocycles
ongenerators}
∈M
N×N
CG
holom.
Thereishop
eto
improvetheanalysisneeded
forexistence
ofstationarysolutions.
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
15/16
Group case.
As a corollary you get:
δ(G ) = β(2)1
(G ) − β(2)0
(G ) − 1
provided that H1(G ; ℓ2(G )) can be generated by a cocycle whose values at thegenerators of G are �analytic� (i.e., are convergent non-commutative power seriesrather than arbitrary elements of ℓ2).
Examples
Freegroups;groupswith
β(2
)1
=0;
free
products,�nite-index
extensions/subgroups,....
Can
probably
handlegroupswhere
P:ℓ2
(G)N
→{values
ofℓ2
cocycles
ongenerators}
∈M
N×N
CG
holom.
Thereishop
eto
improvetheanalysisneeded
forexistence
ofstationarysolutions.
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
15/16
�Deformations�
The lower estimate we get this way on δ is (with few exceptions) the best knownso far.
Examples
(i)
Γin�nite;
∂(x
)=
[x,S
](inner)
=⇒
�deformation�
αt(x
)=
utxu
∗ t;conclusion:
δ≥
1(ii)
Γ=
Γ1∗
Γ2,M
idi�use;∂(x1)
=[x1,S
]forx 1
∈Γ1;∂(x2)
=0forx 2
∈Γ2.
=⇒
�deformation�
αt(x1)
=utx 1u∗ t;αt(x2)
=x 2;conclusion:
δ(X⃗1,X⃗
2)≥
2,
X⃗j⊂
Γj.
(iii)X⃗
free
semicircularsystem
;∂(X
j)=
δ ijS
i=⇒
�deformation�
αtisthePopa
deformationconnectingid
andfree
�ip
(iv)
M=
W∗ (q-deformed
semicircularsystem
).For
|q|<
0.01thisim
plies
δ>
1andso
noCartan,etc.
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
16/16
�Deformations�
The lower estimate we get this way on δ is (with few exceptions) the best knownso far.
Examples
(i)
Γin�nite;
∂(x
)=
[x,S
](inner)
=⇒
�deformation�
αt(x
)=
utxu
∗ t;conclusion:
δ≥
1(ii)
Γ=
Γ1∗
Γ2,M
idi�use;∂(x1)
=[x1,S
]forx 1
∈Γ1;∂(x2)
=0forx 2
∈Γ2.
=⇒
�deformation�
αt(x1)
=utx 1u∗ t;αt(x2)
=x 2;conclusion:
δ(X⃗1,X⃗
2)≥
2,
X⃗j⊂
Γj.
(iii)X⃗
free
semicircularsystem
;∂(X
j)=
δ ijS
i=⇒
�deformation�
αtisthePopa
deformationconnectingid
andfree
�ip
(iv)
M=
W∗ (q-deformed
semicircularsystem
).For
|q|<
0.01thisim
plies
δ>
1andso
noCartan,etc.
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
16/16
�Deformations�
The lower estimate we get this way on δ is (with few exceptions) the best knownso far.
Examples
(i)
Γin�nite;
∂(x
)=
[x,S
](inner)
=⇒
�deformation�
αt(x
)=
utxu
∗ t;conclusion:
δ≥
1(ii)
Γ=
Γ1∗
Γ2,M
idi�use;∂(x1)
=[x1,S
]forx 1
∈Γ1;∂(x2)
=0forx 2
∈Γ2.
=⇒
�deformation�
αt(x1)
=utx 1u∗ t;αt(x2)
=x 2;conclusion:
δ(X⃗1,X⃗
2)≥
2,
X⃗j⊂
Γj.
(iii)X⃗
free
semicircularsystem
;∂(X
j)=
δ ijS
i=⇒
�deformation�
αtisthePopa
deformationconnectingid
andfree
�ip
(iv)
M=
W∗ (q-deformed
semicircularsystem
).For
|q|<
0.01thisim
plies
δ>
1andso
noCartan,etc.
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
16/16
�Deformations�
The lower estimate we get this way on δ is (with few exceptions) the best knownso far.
Examples
(i)
Γin�nite;
∂(x
)=
[x,S
](inner)
=⇒
�deformation�
αt(x
)=
utxu
∗ t;conclusion:
δ≥
1(ii)
Γ=
Γ1∗
Γ2,M
idi�use;∂(x1)
=[x1,S
]forx 1
∈Γ1;∂(x2)
=0forx 2
∈Γ2.
=⇒
�deformation�
αt(x1)
=utx 1u∗ t;αt(x2)
=x 2;conclusion:
δ(X⃗1,X⃗
2)≥
2,
X⃗j⊂
Γj.
(iii)X⃗
free
semicircularsystem
;∂(X
j)=
δ ijS
i=⇒
�deformation�
αtisthePopa
deformationconnectingid
andfree
�ip
(iv)
M=
W∗ (q-deformed
semicircularsystem
).For
|q|<
0.01thisim
plies
δ>
1andso
noCartan,etc.
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
16/16
�Deformations�
The lower estimate we get this way on δ is (with few exceptions) the best knownso far.
Examples
(i)
Γin�nite;
∂(x
)=
[x,S
](inner)
=⇒
�deformation�
αt(x
)=
utxu
∗ t;conclusion:
δ≥
1(ii)
Γ=
Γ1∗
Γ2,M
idi�use;∂(x1)
=[x1,S
]forx 1
∈Γ1;∂(x2)
=0forx 2
∈Γ2.
=⇒
�deformation�
αt(x1)
=utx 1u∗ t;αt(x2)
=x 2;conclusion:
δ(X⃗1,X⃗
2)≥
2,
X⃗j⊂
Γj.
(iii)X⃗
free
semicircularsystem
;∂(X
j)=
δ ijS
i=⇒
�deformation�
αtisthePopa
deformationconnectingid
andfree
�ip
(iv)
M=
W∗ (q-deformed
semicircularsystem
).For
|q|<
0.01thisim
plies
δ>
1andso
noCartan,etc.
Dim
aShlyakhtenko
(UCLA)
Someappicationsoffreestochastic
calculusto
II1factors.
16/16
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