Freely Markovian Dilations, free SDEs and applications to ... · Overview 1 General dilation results. Dilations of Markovian semigroups Properties useful for Deformation/Rigidity

Freely Markovian Dilations, free SDEs andapplications to Deformation/Rigidity Theory.

Yoann Dabrowski

University of California at Los Angeles and Universite Paris-Est

IHP, may 2011

Yoann Dabrowski Freely Markovian dilations

Overview

1 General dilation results.

• Dilations of Markovian semigroups• Properties useful for Deformation/Rigidity Theory• More free probabilistic properties

2 An easy example : freely markovian dilations coming fromgroup cocycles

• Parthasarathy-Schmidt dilations : non-stochastic andstochastic

• Free analogues• Motivation and ideas for the general case


1. General dilation results : Reminder on Markovsemigroups

(M, τ) finite von Neumann algebra, τ faithful normal trace.Markov semigroup: φt : M → M σ-weakly pointwisecontinuous semigroup (φ0 = id , φt+s = φtφs) of completelypositive maps which are tracial (τ ◦ φt = τ) and unital.We say φt is symmetric if τ(bφt(a)) = τ(φt(b)a).Differential Calculus for Markov semigroups : H M-Mbimodule with antilinear isometry JA densely defined closed operator δ : L2(M, τ)→ H is a realderivation if∀x , y ∈ D(δ) ∩M, δ(xy) = xδ(y) + δ(x)y , δ(x∗) = Jδ(x)

Theorem (Cipriani-Sauvageot)

δ real derivation ⇔ φt = e−t∆/2 symmetric Markov semigroup ofgenerator ∆ = δ∗δ (divergence form operator)

We will assume H has a binormal action (not given in ⇐).Yoann Dabrowski Freely Markovian dilations

1. General dilation results : Reminder on Markovsemigroups

(M, τ) finite von Neumann algebra, τ faithful normal trace.Markov semigroup: φt : M → M σ-weakly pointwisecontinuous semigroup (φ0 = id , φt+s = φtφs) of completelypositive maps which are tracial (τ ◦ φt = τ) and unital.We say φt is symmetric if τ(bφt(a)) = τ(φt(b)a).Differential Calculus for Markov semigroups : H M-Mbimodule with antilinear isometry JA densely defined closed operator δ : L2(M, τ)→ H is a realderivation if∀x , y ∈ D(δ) ∩M, δ(xy) = xδ(y) + δ(x)y , δ(x∗) = Jδ(x)

Theorem (Cipriani-Sauvageot)

δ real derivation ⇔ φt = e−t∆/2 symmetric Markov semigroup ofgenerator ∆ = δ∗δ (divergence form operator)

We will assume H has a binormal action (not given in ⇐).Yoann Dabrowski Freely Markovian dilations

1. General dilation results : Dilation Problem

Dilation Problem : Find a finite (M, τ) ⊃ (M, τ) andαt : M → M trace preserving ∗-automorphism such that∀x ∈ M,EM(αt(x)) = φt(x).

Remark : there are semigroups without dilations([Kummerer-Haagerup-Musat]), and without the tracialityrequirement on M, a result of Sauvageot gives normaldilations.

Remark 2 : there is a general inductive limit argumentenabling to go from ∗-endomorphism to ∗-automorphism(I may concentrate on building ∗-endomorphisms in part 2).


1. General dilation results : Dilation Problem

Dilation Problem : Find a finite (M, τ) ⊃ (M, τ) andαt : M → M trace preserving ∗-automorphism such that∀x ∈ M,EM(αt(x)) = φt(x).

Remark : there are semigroups without dilations([Kummerer-Haagerup-Musat]), and without the tracialityrequirement on M, a result of Sauvageot gives normaldilations.

Remark 2 : there is a general inductive limit argumentenabling to go from ∗-endomorphism to ∗-automorphism(I may concentrate on building ∗-endomorphisms in part 2).


1. General dilation results

Theorem (D.)

Let φt be a Markov semigroup on M and assume either

1. φt is symmetricor

2. φt = e−tA/2 is associated to a non-symmetric Dirichlet form(in the sense of Guido-Isola-Scarletti) with generator A withδ = A− A∗ a derivation L2(M)→ L2(M),∆ = (A + A∗)/2 = δ∗δ for a derivation into a binormalbimodule and D(A∗) ∩ D(A) ∩M a core for ∆1/2.

Then φt has a dilation, i.e. αt trace preserving ∗-automorphism ofM such that ∀x ∈ M,EM(αt(x)) = φt(x).

Rmk 1: [Junge-Ricard-Shlyakhtenko] found an easier proof in thesymmetric case.Rmk 2: The non-symmetric case is useful for free SDE results.


1.2 Deformations for Popa’s Deformation/Rigidity Theory

Remarks:

- semigroups of c.p. maps φt or automorphisms αt are thoughtof as deformations of M

- φt = e−tδ∗δ/2 were first considered by Jesse Peterson.

- Having automorphisms (with additional properties) is moreconvenient

Theorem

1 If φt is symmetric, αt is a malleable deformation in the senseof Popa, i.e. there exists β : M → M automorphism withβ2 = id s.t. αt = βα−tβ.

2 If φt = e−tδ∗δ/2 with δ : L2(M, τ)→ H, we have transfer of

properties as M −M bimodule toK = L2(W ∗(αt(M),M)) L2(M) for our dilation αt .

if H ⊂ (L2(M ⊗Mop))⊕∞ then K ⊂ (L2(M ⊗Mop))⊕∞.if H compact (relative to a regular B and φt |B = idB) so is K.



Remarks:

- semigroups of c.p. maps φt or automorphisms αt are thoughtof as deformations of M

- φt = e−tδ∗δ/2 were first considered by Jesse Peterson.

- Having automorphisms (with additional properties) is moreconvenient

Theorem

1 If φt is symmetric, αt is a malleable deformation in the senseof Popa, i.e. there exists β : M → M automorphism withβ2 = id s.t. αt = βα−tβ.

2 If φt = e−tδ∗δ/2 with δ : L2(M, τ)→ H, we have transfer of

properties as M −M bimodule toK = L2(W ∗(αt(M),M)) L2(M) for our dilation αt .

if H ⊂ (L2(M ⊗Mop))⊕∞ then K ⊂ (L2(M ⊗Mop))⊕∞.if H compact (relative to a regular B and φt |B = idB) so is K.



Remarks:

There is actually a canonical description ofK = L2(W ∗(αt(M),M)) L2(M) from H = H(M, η) for acompletely positive map η : M → Mm(M), there is an explicitcanonical bimodule mapK →

∫ ⊕[0,t] H(W ∗(M, αs(M)), η ◦ Eαs(M))dLeb(s)

For most applications to deformation/rigidity, knowing theM −M bimodule generated by (αt − EM ◦ αt)(M) is enough.It is included in

∫ ⊕[0,t] H(M, φs)⊗M H⊗M H(M, φs)dLeb(s).

There are applications in conjunction with Popa-Ozawa (e.g.For Γ countable discrete group with c.m.a.p and positive firstL2 Betti number L(Γ) has no Cartan subalgebra).


1.3 More free probabilistic Properties

Rmk : Those dilations are always freely Markovian, i.e.W ∗(αs(M), 0 ≤ s ≤ t) and W ∗(αs(M), s ≥ t) are free withamalgamation over W ∗(αt(M))



Theorem (D.)

If δ : M → (L2(M ⊗Mop))n (and M has separable predual), thenthere exists a finite (M, τ) ⊃ (W ∗(αs(M), s ≥ 0), τ) and

containing a free Brownian motion St = (S(1)t , ...,S

(n)t ) free with

M such that for any X ∈ D(δ∗δ),

αt(X ) = α0(X )− 1

2

∫ t

0αs(δ∗δ(X ))ds +

∫ t

0αs ⊗ αs(δ(X ))#dSs .

Rmk 1 : This enables to get a free Transportation cost inequality.Rmk 2: Under stronger Lipschitz like assumptions on δ∗δ(X ) andcoassociativity like assumptions on δ, one can get M = M ∗ L(F∞)and compute microstates free entropy dimension, e.g.



Theorem (D.)

If δ : M → (L2(M ⊗Mop))n (and M has separable predual), thenthere exists a finite (M, τ) ⊃ (W ∗(αs(M), s ≥ 0), τ) and

containing a free Brownian motion St = (S(1)t , ...,S

(n)t ) free with

M such that for any X ∈ D(δ∗δ),

αt(X ) = α0(X )− 1

2

∫ t


∫ t


Rmk 1 : This enables to get a free Transportation cost inequality.Rmk 2: Under stronger Lipschitz like assumptions on δ∗δ(X ) andcoassociativity like assumptions on δ, one can get M = M ∗ L(F∞)and compute microstates free entropy dimension, e.g.



Theorem (D.)

If X1, ...,XN are q-Gaussian variables and |q|N < 1, |q|√

N ≤ 0.13then δ0(X1, ...,XN) = N.


Overview

1 General dilations results.

• Dilations of Markovian semigroups• Properties useful for Deformation/Rigidity Theory• More free probabilistic properties

2 An easy example : freely markovian dilations coming fromgroup cocycles

• Parthasarathy-Schmidt dilations :“non-stochastic” andstochastic

• Free analogues• Motivation and ideas for the general case


2.1 Parthasarathy-Schmidt dilations :“non-stochastic”variant

Consider π : Γ→ O(H)) orthogonal representation of Γ(countable discrete group)

If (Y , ν) =∏

n≥1(IR, 1√2π

e−x2/2dx) Xn n-th coordinate

Gaussian r. v. (L2(Y , ν) ' Fσ(`2) symmetric Fock space).

For {en} ⊂ H ort. basis define isometry (”Gaussian functor”)

G (∑n≥1

cnen) =∑n≥1

cnXn

Γ y L∞(Y ) by g .ω(ξ) = ω(π(g)ξ) for ω(ξ) = exp(iG (ξ))

For b : Γ→ H cocycle (i.e. b(gh) = π(g)b(h) + b(g))Get a dilation of φt(ug ) = exp(−t2||b(g)||2/2)ug , αt definedon L∞(Y ) o Γ

αt(ug ) = ω(tb(g))ug , αt |L∞(Y ) = id




If (Y , ν) =∏

n≥1(IR, 1√2π




G (∑n≥1

cnen) =∑n≥1

cnXn







If (Y , ν) =∏

n≥1(IR, 1√2π




G (∑n≥1

cnen) =∑n≥1

cnXn





2.1 Parthasarathy-Schmidt dilations : stochastic variant

Replace (H, π) by (H = H⊗ L2(IR+, Leb), π ⊗ id),

Consider G gaussian functor given for {en} basis of HFor b : Γ→ H cocycleGet a dilation of φt(g) = exp(−t||b(g)||2/2),αt defined on L∞(Y ) o Γ in order to get αtαs = αt+s

αt(ug ) = ω(b(g)⊗1[0,t))ug , αt(ω(ξ⊗1[0,s))) = ω(ξ⊗1[t,t+s))

Bt(fi ) = G (fi ⊗ 1[0,t)) independent Brownian motions for {fi}orthonormal basis of HBest viewpoint on ωt(g) = exp(iB(b(g)⊗ 1[0,t))) is assolution of a linear SDE :ωt(g) = 1− 1

2

∫ t0 ds||b(g)||2ωs(g) + i

∫ t0 dBs(b(g))ωs(g)

Rmk: - easy to solve by Picard iteration,- Best viewpoint in free case because of a different Ito formula







2

∫ t0 ds||b(g)||2ωs(g) + i









2

∫ t0 ds||b(g)||2ωs(g) + i









2

∫ t0 ds||b(g)||2ωs(g) + i




2.2 Free analogue of the stochastic variant

Keep (H = H⊗ L2(IR+, Leb), π ⊗ id),

Consider S free gaussian functor given for {en} basis of H i.e.S : H → L2(L(F∞), τ) ' F(H) ' L2(?n≥1W ∗(Sn)),Sn semicircular elements

S(∑n≥1

cnen) =∑n≥1

cnSn.

St(fi ) = S(fi ⊗ 1[0,t)) are free Brownian motions.

For b : Γ→ H cocycle, Solve

wt(g) = 1− 1

2

∫ t

0ds||b(g)||2ws(g) + i

∫ t

0dSs(b(g))ws(g)

One can check wt(g) is a unitary.Pb: Find a twisted action Γ y ?n≥1W ∗(Sn) makingαs(ug ) = ws(g)ug an homomorphism.





S(∑n≥1

cnen) =∑n≥1

cnSn.



wt(g) = 1− 1

2

∫ t

0ds||b(g)||2ws(g) + i

∫ t

0dSs(b(g))ws(g)






S(∑n≥1

cnen) =∑n≥1

cnSn.



wt(g) = 1− 1

2

∫ t

0ds||b(g)||2ws(g) + i

∫ t

0dSs(b(g))ws(g)




Answer : there is a natural choice.We want wt(gh) = wt(g)ugwt(h)u∗g First order in SDE :

dSs(b(gh))ws(gh) 'dSs(b(g))ws(g)gws(h)g−1 + ws(g)ugdSs(b(h))u∗gugws(h)u∗g

But we know from the cocycle propertydSs(b(gh))ws(gh) = dSs(b(g))ws(gh)+dSs(π(g).b(h))ws(gh)This suggests ugdSs(b(h))u∗g = ws(g)∗dSs(π(g).b(h))ws(g).Note the twist by ws(g) disappears in classical Brownian casebecause of commutation. Note the action givesαs(ug )dSs(b(h))αs(ug )∗ = dSs(π(g).b(h)) usual freeGaussian action.In correct order, define ws(g), check we get a trace p.homomorphism “α(g) = ug .u

∗g” on W ∗(S(gb(h)⊗ 1[0,t))′s),

check wt(gh) = wt(g)α(g)(wt(h)), check one gets an actionof Γ.






∗g” on W ∗(S(gb(h)⊗ 1[0,t))′s),







∗g” on W ∗(S(gb(h)⊗ 1[0,t))′s),







∗g” on W ∗(S(gb(h)⊗ 1[0,t))′s),



2.3 Motivation for the general case

At the end,

αt(ug ) = ug −1

2

∫ t

0ds||b(g)||2αs(ug ) + i

∫ t

0dSs(b(g))αs(ug ).

Consider the case H = `2(γ) the regular representation.One can check the action on∫ t

0 αs(h)dSs(e)αs(h)∗ = S(g ⊗ 1[0,t)), and Ss(e) = Ss(δe) is a freebrownian motion free from Γ in W ∗(S ′ns) o Γ.

αt(ug ) = ug −1

2

∫ t

0dsαs(δ∗δ(ug )) +

∫ t

0αs ⊗ αs(δ(ug ))#dSs(e)

where δ(ug ) = i∑

h bh(g)uh ⊗ u∗hug is the natural real derivationwith value L2(L(Γ)⊗ L(Γ)op) with δ∗δ(ug ) = ||b(g)||2ug , anda⊗ b#Ss = aSsb.



At the end,

αt(ug ) = ug −1

2

∫ t

0ds||b(g)||2αs(ug ) + i

∫ t

0dSs(b(g))αs(ug ).



αt(ug ) = ug −1

2

∫ t


∫ t






At the end,

αt(ug ) = ug −1

2

∫ t

0ds||b(g)||2αs(ug ) + i

∫ t

0dSs(b(g))αs(ug ).



αt(ug ) = ug −1

2

∫ t


∫ t






This suggests the general equation in the case of derivations notcoming from group cocyles :

αt(X ) = α0(X )− 1

2

∫ t


∫ t


This equation was first considered by D. Shlyakhtenko also for thefree difference quotient δ.

My method of resolution is a “Path space approach”. We deducefrom Ito formula equations for τ |W ∗(αt1(M), ..., αtn(M)) to buildfirst τ on a universal free product C ∗ algebra.



This suggests the general equation in the case of derivations notcoming from group cocyles :

αt(X ) = α0(X )− 1

2

∫ t


∫ t


This equation was first considered by D. Shlyakhtenko also for thefree difference quotient δ.

My method of resolution is a “Path space approach”. We deducefrom Ito formula equations for τ |W ∗(αt1(M), ..., αtn(M)) to buildfirst τ on a universal free product C ∗ algebra.


Conclusion

Hope for more applications to Deformation/Rigidity Theory

Better understand the free probabilistic structure of M (e.g.for more free entropy dimension computations).

Solve non-stationary free SDEs.

Thank you for your attention.


Conclusion

Hope for more applications to Deformation/Rigidity Theory

Better understand the free probabilistic structure of M (e.g.for more free entropy dimension computations).

Solve non-stationary free SDEs.

Thank you for your attention.


Documents

Freely Markovian Dilations, free SDEs and applications to ... · Overview 1 General dilation results. Dilations of Markovian semigroups Properties useful for Deformation/Rigidity