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Some appications of free stochastic calculus to II1factors.

Dima Shlyakhtenko

UCLA

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