Dilation theory and applications Marius Junge Joint with Eric - - PowerPoint PPT Presentation

Dilation theory and applications

Marius Junge Joint with Eric Ricard and Dima Shlyakhtenko Workshop: II1 factors: rigidity, symmetries and classification

UI at Urbana-Champaign

• 27. Mai 2011

Marius Junge (UI at Urbana-Champaign) Dilation

• 27. Mai 2011

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Plan

Background and examples Review: Stinespring and modules Markov dilation Dilation for single maps Main results Technical condition Gradient algebra Application(’s) (Avsec)

Marius Junge (UI at Urbana-Champaign) Dilation

• 27. Mai 2011

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Setup and Motivation

➩ In the following I am interested in a pair (N, Tt) where N is a finite von Neumann algebra and Tt a semigroup of completely positive, unital selfadjoint maps, i.e. τ(Tt(x)y) = τ(xTt(y)). ➩ Example: N = L∞(Td) = L(Zd) and Tt(λ(k)) = e−t|k|αλ(k) for 0 < α ≤ 2. ➩ Examples: N = L(Rd

discrete) and Tt(λ(k)) = e−t|k|αλ(k) for

0 < α ≤ 2 (Bohr compactification). ➩ Example: N = Γq(ℓd

2) Bozejko-Speichers’s q-gaussian factors, and

Tt = e−tN is given by the number operator. ➩ Example: N = L∞(K) ⋊ Γ where K is compact Riemanian manifold, Tt = e−t∆ given by the Laplace Beltrami operator, and Γ a discrete group of diffeomorphisms.

Marius Junge (UI at Urbana-Champaign) Dilation

• 27. Mai 2011

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Stinespring and modules

➤ We fill first consider a fixed a completely positive unital normal and selfadjoint map T. ➤ By GNS construction N ⊗T N with inner product a ⊗ b, c ⊗ d = b∗T(a∗c)d . ➤ Theorem: (Stinespring-Kasparov-Paschke) There exists π : N → B(ℓ2)¯ ⊗N such that T(x) = e11π(x)e11 . ➤ Drawback: π is no longer trace preserving

Marius Junge (UI at Urbana-Champaign) Dilation

• 27. Mai 2011

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Markov Dilation

☞ A Markov dilation for T is given by trace preserving

∗-homomorphisms πk : N → Nk ⊂ N in a bigger finite von Neuamnn

algebra N with increasing filtration Nk such that N

T j−k

→ N ↓πj ↓πk N

ENk

→ Nk ☞ Theorem (K¨ ummerer) (Haagerup-Musat) There are s.a. ucp maps in matrix algebras which do not admit a Markov dilation. ☞ Haagerup-Musat’s work is based on Anantharaman Delaroche’s notion of fatorizable maps. ☞ Non-factorizable maps (j = 1, k = 0) can be reconstructed from K¨ ummerer’s work. ☞ In the commutative case all cpu maps have a Markov dilation.

Marius Junge (UI at Urbana-Champaign) Dilation

• 27. Mai 2011

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Folklore examples I

➯ N = L∞(Ω, Σ, µ) the Markov dilation is obtained from constructing a measure on Ω × Ω[0,∞) by the so called Kolmogorov construction. ➯ N = L∞(Rd, λ), λ Lebesgue measure Px is the Wiener measure on continuous path, µ(A) =

• Px(Ax)dx and

πt(f )(ω, x) = f (Bt + x) , where Bt = (B1

t , ..., Bd t ) is given by an orthogonal family of brownian

motions. ➯ b : G → Rd, d ∈ N ∪ {∞} be a cocycle. Let L∞(ΩRd, µ) be given by the gaussian measure space construction, i.e the commutative vNa generated by the brownian motion. Then πt(λ(g)) = exp(iBt(b(g))) λ(g) gives a Markov dilation for Tt(λ(g)) = e−tb(g)2λ(g).

Marius Junge (UI at Urbana-Champaign) Dilation

• 27. Mai 2011

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Folklore examples II

➯ For a Riemann manifold (embedded in Rn) one has to construct a brownian motion so that tangent directions remain in the tangent

• space. Well-known “rolling without slipping in probability theory”.

➯ Let ak ∈ N be selfadjoint random variables and Bk

t independent

brownian motions. Let Bt =

n

• k=1

Bk

t ak and α = k a2

• k. Then the

solution to the SDE dut = −α 2 utdt + idBtut u0 = 1 is a unitary and satisfies Es(u∗

t xut) = u∗ s (Tt−s(x))us

where Tt = exp(−tA) and the generator is given by A(x) =

• k

(a2

kx + xa2 k − 2akxak) .

Marius Junge (UI at Urbana-Champaign) Dilation

• 27. Mai 2011

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Remark: The existence and uniqueness of the SDE is proved as in any

• rdinary ODE class using the Banach contraction principle.

Corollary: (K¨ ummerer-Maassen) Every semigroup of selfadjoint cpu maps

• n Matrix algebras admits a continuous Markov dilation

Es(πt(x)) = πs(Tt−s(x)) with values in L∞(Ω, Σ, µ; Mn).

Marius Junge (UI at Urbana-Champaign) Dilation

• 27. Mai 2011

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Back to a single T: Fock space

➩ Let T : N → N be a normal cpu. Then F =

∞

• n=0

(N ⊗T N)⊗n is a N-bimodule (Pimsner, Speicher). ➩ Lemma: There exists a conditional expectation E : L(F) → N and an element ξ ∈ L(F) such that T(x) = E(ξ∗π(x)ξ) . ➩ Key Observation: (Shlyakhtenko) If T is selfadjoint the algebra generated by ξ and N is again finite and E becomes a trace preserving conditional expectation.

Marius Junge (UI at Urbana-Champaign) Dilation

• 27. Mai 2011

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➩ Modification: Similarly we may consider, F([0, ∞)) =

∞

• n=0

L2(0, ∞; N ⊗T N)⊗n as a N bimodule. Then the von Neumann algebra generated by bt = l(1[0,t] ⊗ 1 ⊗ 1) + l∗(1[0,t] ⊗ 1 ⊗ 1) and N is finite and (bt)t ≥ 0 is a free brownian motion such that E(btxbt) = tT(x) . ➩ Proposition: The solution to dut = −1 2utdt + idbtut u0 = 1 is a unitary such that St(x) = exp(−t(I − T))x satisfies Es(u∗

t xut) = u∗ s St−s(x)us .

Marius Junge (UI at Urbana-Champaign) Dilation

• 27. Mai 2011

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Dilation problem

➩ The solution ut is in the algebra generated by the bt. Indeed in the Fock space we have ut ∼ = e−t/2

∞

• k=0

ik1[0,t]k ⊗ (1 ⊗ 1)k ⊗ 1 . ➩ Theorem: Let Tt be a semigroup of unital completely positive normal selfadjoint maps, then Tt admits a Markov dilation, i.e. there exists an increasing filtration Ns in a finite von Neumann N and πs : N → Ns such that Et(πs(x)) = πt(Ts−tx) t < s . ➩ Proof: We use an ultraproduct construction T ω

t = (e−th−1(I−Th))• h>0

and observe that N ⊂ Nω is an invariant subspace.

Marius Junge (UI at Urbana-Champaign) Dilation

• 27. Mai 2011

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Regularity property

➫ In applications to harmonic analysis we also need a reversed dilation πs : N → N[s,∞] ⊂ N such that N[s,∞] are decreasing in s and E[t,∞](πs(x)) = πt(Ts−tx) . ➫ We say that a reversed Markov dilation is a.u. continuous if if for every ε > 0 and t0 > 0 there exists e ∈ N with τ(1 − e) < ε and s → πs(Tsx)e is norm continuous on [0, t0].

Marius Junge (UI at Urbana-Champaign) Dilation

• 27. Mai 2011

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Regularity

➮ Theorem: Let Tt a semigroup of completely positive unital selfadjoint maps. The following are equivalent: i) Tt admits a reversed Markov dilation which is a.u. continuous; ii) Tt = e−tA admits a derivation for A with values in a normal N bimodule; iii) For every x ∈ dom(A1/2) the gradient 2Γ(x, x) = A∗(x)x + x∗A(x) − A(x∗x) = lim

h→0

x∗x + Th(x∗x) − Th(x∗)x − x∗Th(x) h ∈ L1(N) . ➮ Theorem: There are semigroups on a commutative vNa which violate this condition. ➮ Dabrowski has obtained the same dilation theorem under the additional condition ii) but for larger class of semigroups.

Marius Junge (UI at Urbana-Champaign) Dilation

• 27. Mai 2011

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Gradient module-Part 1

Following the work of Sauvageot we may define on B = dom(A1/2) ⊗ N the B∗-valued inner product a ⊗ b, c ⊗ dΓ = a∗Γ(b, c)d ∈ dom(A1/2)∗ . There derivation is given by δ(a) = a ⊗ 1 − 1 ⊗ a . The corresponding scalar inner product is given by (ξ, η) = ξ, η(1) with completion H = L2(Γ).

Remark

Our results shows that the right action is normal iff Γ(b, c) ∈ L1(N) for all a, b ∈ dom(A1/2).

Marius Junge (UI at Urbana-Champaign) Dilation

• 27. Mai 2011

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Free Gradient module

➮ Proposition: There exists a finite von Neumann algebra N(Γ) containing N and a derivation δ : dom(A1/2) → N(Γ) such that E(δ(x)∗δ(y)) = Γ(x, y) . and N(Γ) is generated by δ(N) and N. N(Γ) is an example of Shlyakhtenko’s N valued semicircular systems obtained from Γ. ➮ Examples: N = L∞(Rd) and Tt heat semigroup, then N = L∞(Fd) ⊗ L∞(Rd), the free analogue of the Markov dilation given by the gaussian measure motion. Recall L∞(Fd) = Γ0(ℓd

2) is

generated by d free semicircular random variables. ➮ Examples: b : G → Rd cocycle. Then N = Γ0(ℓd

2) ⋊ G.

➮ Remark: N(Γ) is the free analogue of L∞(T ∗X) for a manifold X, T ∗X the cotangent bundle. It is open whether N(Γ) admits a geodesic flow-that would give perfect intrinsic deformation!

Marius Junge (UI at Urbana-Champaign) Dilation

• 27. Mai 2011

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Weak containement

➮ Let T : N → N be a cpu map. Then we write T L2(N ⊗ N) if L2(N ⊗T N) ⊖ L2(N) L2(N ⊗ N). ➮ Recall that a Hilbert N bimodule H L2(N ⊗ N) if for every ξ = 1 the functional ϕξ(a ⊗ b) = (ξ, aξb) is in (N ⊗min Nop)∗. ➮ Let N = LG and ∆(λ(g)) = λ(g) ⊗ λ(g). For a Fourier (Herz-Schur) multiplier Tt(x) = (ϕt ⊗ id)∆(x) we have ϕt ∈ LG ∗ ⇒ Tt L2(N ⊗ N) . ➮ Note that for abelian G there are recent results showing ϕt ∈ L1(LG).

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• 27. Mai 2011

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For G = Fn, N = LFn and the length function Tt(λ(g)) = e−t|g|λ(g) we have dt ∈ L2(LG) for t > log n, but L2(Γ) L2(N ⊗ N) and Tt L2(N ⊗ N).

Marius Junge (UI at Urbana-Champaign) Dilation

• 27. Mai 2011

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An alternative

Theorem

Let Tt be a semigroup of selfadjoint ucp maps such that Tt L2(N ⊗ N) for all t > 0 or L2(Γ) L2(N ⊗ N). Assume that N has the w∗ CBAP. Let P ⊂ N be amenable and U(P) ⊂ G ⊂ NA. Then ∃z∈Z(G ′∩N) limt→0 supx∈zP,x≤1 Tt(x) − x2 = 0 ր

• r

ց G ′′ ∩ N ameanable Here NA ⊂ N is the relative normalizer. Builds on work of: Dabrowski, Houdayer-Shlyakhtenko, Ozawa-Popa

Marius Junge (UI at Urbana-Champaign) Dilation

• 27. Mai 2011

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Idea and “application”

In the proof we show ∀tTt L2(N ⊗ N) ⇒ L2(Γ) L2(N ⊗ N) and moreover, if the gradient module is weakly contained in the coarse representation and the regularity assumption is satisfied, then vNa N given by the Markov dilation also satisfies NL2(N)M ⊖ L2(N) L2(N ⊗ N) using a martingale representation theorem. That is enough to fuel Popa-Ozawa machinery (driven ´ a la Houdayer-Shlyakhtenko).

Theorem (Steve Avsec)

Let Γq(ℓd

2) the II1 factor generated by d orthogonal q-gaussian random

variables, and d ≤ d(q). Then Γq(ℓd

2) is strongly solid for −1 < q < 1.

Marius Junge (UI at Urbana-Champaign) Dilation

• 27. Mai 2011

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Open problems

1 Construct a geodesic flow for (N, Tt). 2 Is it true that Tt L2(N ⊗ N) iff L2(Γ) L2(N ⊗ N)? This is true in

the “flat” situation for cocycles. Thanks for listening

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