Dilation theory and applications Marius Junge Joint with Eric Ricard and Dima Shlyakhtenko Workshop: II 1 factors: rigidity, symmetries and classification UI at Urbana-Champaign 27. Mai 2011 Marius Junge (UI at Urbana-Champaign) Dilation 27. Mai 2011 1 / 20
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 2 / 20
Setup and Motivation ➩ In the following I am interested in a pair ( N , T t ) where N is a finite von Neumann algebra and T t a semigroup of completely positive, unital selfadjoint maps, i.e. τ ( T t ( x ) y ) = τ ( xT t ( y )). ➩ Example: N = L ∞ ( T d ) = L ( Z d ) and T t ( λ ( k )) = e − t | k | α λ ( k ) for 0 < α ≤ 2. discrete ) and T t ( λ ( k )) = e − t | k | α λ ( k ) for ➩ Examples: N = L ( R d 0 < α ≤ 2 (Bohr compactification). ➩ Example: N = Γ q ( ℓ d 2 ) Bozejko-Speichers’s q -gaussian factors, and T t = e − tN is given by the number operator. ➩ Example: N = L ∞ ( K ) ⋊ Γ where K is compact Riemanian manifold, T t = 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 3 / 20
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 ( ℓ 2 )¯ ⊗ N such that T ( x ) = e 11 π ( x ) e 11 . ➤ Drawback: π is no longer trace preserving Marius Junge (UI at Urbana-Champaign) Dilation 27. Mai 2011 4 / 20
Markov Dilation ☞ A Markov dilation for T is given by trace preserving ∗ -homomorphisms π k : N → N k ⊂ N in a bigger finite von Neuamnn algebra N with increasing filtration N k such that T j − k → N N ↓ π j ↓ π k E N k N → N k ☞ 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 5 / 20
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 ∞ ( R d , λ ), λ Lebesgue measure P x is the Wiener measure on � continuous path, µ ( A ) = P x ( A x ) dx and π t ( f )( ω, x ) = f ( B t + x ) , where B t = ( B 1 t , ..., B d t ) is given by an orthogonal family of brownian motions. ➯ b : G → R d , d ∈ N ∪ {∞} be a cocycle. Let L ∞ (Ω R d , µ ) be given by the gaussian measure space construction, i.e the commutative vNa generated by the brownian motion. Then π t ( λ ( g )) = exp( iB t ( b ( g ))) λ ( g ) gives a Markov dilation for T t ( λ ( g )) = e − t � b ( g ) � 2 λ ( g ). Marius Junge (UI at Urbana-Champaign) Dilation 27. Mai 2011 6 / 20
Folklore examples II ➯ For a Riemann manifold (embedded in R n ) 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 a k ∈ N be selfadjoint random variables and B k t independent n B k k a 2 brownian motions. Let B t = � t a k and α = � k . Then the k =1 solution to the SDE du t = − α 2 u t dt + idB t u t u 0 = 1 is a unitary and satisfies E s ( u ∗ t xu t ) = u ∗ s ( T t − s ( x )) u s where T t = exp ( − tA ) and the generator is given by � ( a 2 k x + xa 2 k − 2 a k xa k ) . A ( x ) = k Marius Junge (UI at Urbana-Champaign) Dilation 27. Mai 2011 7 / 20
Remark: The existence and uniqueness of the SDE is proved as in any ordinary ODE class using the Banach contraction principle. Corollary: (K¨ ummerer-Maassen) Every semigroup of selfadjoint cpu maps on Matrix algebras admits a continuous Markov dilation E s ( π t ( x )) = π s ( T t − s ( x )) with values in L ∞ (Ω , Σ , µ ; M n ). Marius Junge (UI at Urbana-Champaign) Dilation 27. Mai 2011 8 / 20
Back to a single T : Fock space ➩ Let T : N → N be a normal cpu. Then ∞ � ( N ⊗ T N ) ⊗ n F = n =0 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 9 / 20
➩ Modification: Similarly we may consider, ∞ L 2 (0 , ∞ ; N ⊗ T N ) ⊗ n � F ([0 , ∞ )) = n =0 as a N bimodule. Then the von Neumann algebra generated by b t = l (1 [0 , t ] ⊗ 1 ⊗ 1) + l ∗ (1 [0 , t ] ⊗ 1 ⊗ 1) and N is finite and ( b t ) t ≥ 0 is a free brownian motion such that E ( b t xb t ) = tT ( x ) . ➩ Proposition: The solution to du t = − 1 2 u t dt + idb t u t u 0 = 1 is a unitary such that S t ( x ) = exp( − t ( I − T )) x satisfies E s ( u ∗ t xu t ) = u ∗ s S t − s ( x ) u s . Marius Junge (UI at Urbana-Champaign) Dilation 27. Mai 2011 10 / 20
Dilation problem ➩ The solution u t is in the algebra generated by the b t . Indeed in the Fock space we have ∞ i k 1 [0 , t ] k ⊗ (1 ⊗ 1) k ⊗ 1 . u t ∼ = e − t / 2 � k =0 ➩ Theorem: Let T t be a semigroup of unital completely positive normal selfadjoint maps, then T t admits a Markov dilation, i.e. there exists an increasing filtration N s in a finite von Neumann N and π s : N → N s such that E t ( π s ( x )) = π t ( T s − t x ) t < s . t = ( e − th − 1 ( I − T h ) ) • ➩ Proof: We use an ultraproduct construction T ω h > 0 and observe that N ⊂ N ω is an invariant subspace. Marius Junge (UI at Urbana-Champaign) Dilation 27. Mai 2011 11 / 20
Regularity property ➫ In applications to harmonic analysis we also need a reversed dilation π s : N → N [ s , ∞ ] ⊂ N such that N [ s , ∞ ] are decreasing in s and E [ t , ∞ ] ( π s ( x )) = π t ( T s − t x ) . ➫ We say that a reversed Markov dilation is a.u. continuous if if for every ε > 0 and t 0 > 0 there exists e ∈ N with τ (1 − e ) < ε and s �→ π s ( T s x ) e is norm continuous on [0 , t 0 ]. Marius Junge (UI at Urbana-Champaign) Dilation 27. Mai 2011 12 / 20
Regularity ➮ Theorem: Let T t a semigroup of completely positive unital selfadjoint maps. The following are equivalent: i) T t admits a reversed Markov dilation which is a.u. continuous; ii) T t = e − tA admits a derivation for A with values in a normal N bimodule; iii) For every x ∈ dom( A 1 / 2 ) the gradient 2Γ( x , x ) = A ∗ ( x ) x + x ∗ A ( x ) − A ( x ∗ x ) x ∗ x + T h ( x ∗ x ) − T h ( x ∗ ) x − x ∗ T h ( x ) = lim ∈ L 1 ( N ) . h h → 0 ➮ 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 13 / 20
Gradient module-Part 1 Following the work of Sauvageot we may define on B = dom( A 1 / 2 ) ⊗ N the B ∗ -valued inner product � a ⊗ b , c ⊗ d � Γ = a ∗ Γ( b , c ) d ∈ dom( A 1 / 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 = L 2 (Γ). Remark Our results shows that the right action is normal iff Γ( b , c ) ∈ L 1 ( N ) for all a , b ∈ dom( A 1 / 2 ) . Marius Junge (UI at Urbana-Champaign) Dilation 27. Mai 2011 14 / 20
Free Gradient module ➮ Proposition: There exists a finite von Neumann algebra N (Γ) containing N and a derivation δ : dom ( A 1 / 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 ∞ ( R d ) and T t heat semigroup, then N = L ∞ ( F d ) ⊗ L ∞ ( R d ), the free analogue of the Markov dilation given by the gaussian measure motion. Recall L ∞ ( F d ) = Γ 0 ( ℓ d 2 ) is generated by d free semicircular random variables. ➮ Examples: b : G → R d cocycle. Then N = Γ 0 ( ℓ d 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 15 / 20
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