Mixed q-gaussian algebras and free transport

Transcription

1 Mixed q-gaussian algebras and free transport Qiang Zeng (Northwestern University) Joint with Marius Junge (Urbana) and Brent Nelson (Berkeley) Free probability and large N limit, V Berkeley, March 2016 Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 1 / 35

2 Outline 1 Mixed q-gaussian algebras Γ Q (R N ) 2 Isomorphism theorem for Γ Q (R N ) 3 Free transport for infinite variables Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 2 / 35

3 The object to study In 1993, Speicher introduced the mixed q-commutation relation. li l j q ij l j li = δ i=j where q ij = q ji [ 1, 1], i, j = 1, 2,..., N. Speicher (1993), Bożejko Speicher (1994) showed that it has Fock space representation: l i, li can be represented as left creation and annihilation operators on a certain Fock space. Γ Q (R N ) := vn(x Q i, i = 1,..., N), XQ i := l i + li, Q := (q ij) 1 i,j N. We call Γ Q (R N ) the mixed q-gaussian algebras. Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 3 / 35

4 Why consider mixed q-gaussian? Some examples: qij q [ 1, 1], the q-gaussian algebra Γ q (R N ). Some special pattern of Q: i Γ qi (R ni ), i (Γ qi (R ni ) Γ pi (R mi )),... Mixed q-commutation relation still verifies the braid relation (or Yang Baxter equation): Let T B(H H), T (e i e j ) = q ij e j e i. Then (T 1)(1 T )(T 1) = (1 T )(T 1)(1 T ). It provides an example to study free transport for infinite variables (extending Guionnet Shlyakhtenko s theorem). Many results for q-gaussian algebras and Bożejko Speicher s algebras: Bożejko, Kümmerer, Speicher, Dykema, Nica, Biane, Krolak, Lust-Piquard, Shlyakhtenko, Nou, Śniady, Ricard, Anshelevich, Belinschi, Lehner, Kennedy, Avsec, Dabrowski, Guionnet, et al. Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 4 / 35

5 Speicher s central limit theorem: Notation J N,m := [N] [m], J N := [N] N. ε : J N J N { 1, 1}, ε(x, y) = ε(y, x), ε(x, x) = 1. A m = alg(x i (k), (i, k) J N,m ), where x i (k) s satisfy x i (k) = x i (k) and x i (k)x j (l) ε((i, k), (j, l))x j (l)x i (k) = 2δ (i,k),(j,l) for (i, k), (j, l) J N,m. A m can be represented as a matrix subalgebra of M 2 Nm. A word of A m x B = x i1 (k 1) x id (k d), B = {(i 1, k 1 ),, (i d, k d )} J N,m. A normalized trace τ m on A m : τ m (x B ) = δ B,. Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 5 / 35

6 Speicher s central limit theorem Consider independent random variables ε((i, k), (j, l)) : Ω { 1, 1} for (i, k) < (j, l) on (Ω, P) with distribution P(ε((i, k), (j, l)) = 1) = 1 q ij 2 (i, k), (j, l) [N] N. x i (m) := 1 m m k=1 x i(k)., P(ε((i, k), (j, l)) = 1) = 1 + q ij, 2 Theorem (Speicher 93) Let i [N] s. Then lim m τ m( x i1 (m) x is (m)) = δ s 2Z σ P 2 (s) σ σ(i) {r,t} I(σ) q(i(e r ), i(e t )) a.s. Here and in what follows, we understand {i,j} q(i, j) = 1. Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 6 / 35

7 Speicher s central limit theorem: ctd σ π or π σ iff σ is a refinement of π. Given i = (i 1,, i d ) [N] d, we associate a partition σ(i) to i by requiring k, l [d] belonging to the same block of σ(i) iff i k = i l. P 2 (d) consists of π = {V 1,, V d/2 } such that V k = 2. Write V k = {e k, z k } with e k < z k and e 1 < e 2 < < e d/2. Given π P 2 (d), the set of crossings of π is I(π) = {{k, l} 1 k, l d/2, e k < e l < z k < z l }. Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 7 / 35

8 Fock space representation Inner product: e i1 e im, e j1 e jn Q = δ m,n σ S n a(σ, j) e i1, e jσ 1 (1) e im, e jσ 1 (n) on CΩ d 1 (RN ) d, and denote the completion by F Q (R N ). l(e n ) denotes the left creation operator l(e n )Ω = e n l(e n )e i1 e id = e n e i1 e id, Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 8 / 35

9 Fock space representation: ctd Its adjoint is given by the left annihilation operator l(e n ) Ω = 0 and d l(e n ) e i1 e id = δ n=ik q ni1 q nik 1 e i1 ê ik e id, k=1 where ê ik means that e ik is omitted in the tensor product. We will also need the right creation operator r(e n ) defined by r(e n )(e i1 e id ) = e i1 e id e n. Bożejko Speicher ( 94) l(e n ) = r(e n ) = { 1 1 qnn if q nn [0, 1), 1 if q nn ( 1, 0]. Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 9 / 35

10 Ultraproduct construction Let U be a free ultrafilter on N. We get a finite vna A U := m,u A m with normal faithful tracial state τ U = lim m,u τ m. A U := p< L p (A U ). For each ω Ω, ( x i (m)(ω)) A U. Here ( x i (m)(ω)) is the element represented by ( x i (m)(ω)) m N in the ultraproduct. Speicher s CLT implies τ U (( x i1 (m)(ω)) ( x is (m)(ω)) ) = δ s 2Z and τ U ( ( x i (m)(ω)) p ) Cp p. σ P 2 (s) σ σ(i) {r,t} I(σ) q(i(e r ), i(e t )) Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 10 / 35

11 Ultraproduct construction: ctd Junge ( 06): the von Neumann algebras generated by the spectral projections of ( x i (m)(ω)), i = 1,, N for different ω Ω are isomorphic. Γ Q (R N ) is any von Neumann algebra in the isomorphic class with generators X Q i := ( x i (m)(ω)), i = 1,, N. X Q i may be unbounded, therefore may not be in Γ Q. But it belongs to Γ Q := p< L p (Γ Q, τ U ). τ Q = τ U ΓQ. Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 11 / 35

12 Wick word decomposition Theorem (Junge-Z, 15) Let ( x j (m)) p< L p ( m,u L (Ω; A m )) for j = 1,, d. Then holds for all ω Ω. ( x i1 (m)) ( x id (m)) = m d/2 σ P 1,2 (d) σ σ(i) w σ (i) ( Here w σ (i) = 1, k [m] d :σ(k)=σ E N s(k)[x i1 (k 1 ) x id (k d )]) Ns (k) denotes the von Neumann algebra generated by all x iα (k α ) s, where k α corresponds to singleton blocks in σ(k). Proof is a bit technical: Based on NC Khintchine and martingale inequalities, decoupling, Pisier s method for multi-index summations (Möbius inversion), etc. Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 12 / 35

13 The Ornstein Uhlenbeck semigroup The Wick word (or Wick product) of e i1 e in,w (e i1 e in ), is the unique element in Γ Q satisfying W (e i1 e in )Ω = e i1 e in. W (e i1 e in ) can be identified with ( 1. w(i) = m s/2 x i1 (k 1 ) x is (k s )) k [m] s :σ(k) P 1 (s) The O-U semigroup is defined by T t (W (e i1 e in )) = e t i W (e i1 e in ). Equivalently, for i [N] s, ( 1 T t w(i) = m s/2 k:σ(k) P 1 (s) e ts x i1 (k 1 ) x is (k s )) = e ts w(i). Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 13 / 35

14 Analytic properties: Hypercontractivity Theorem (Junge Z 15) Let q ij [ 1, 1]. Then for 1 p, r <, T t Lp L r = 1 if and only if e 2t p 1 r 1. Let A be the generator of T t. Meyers carr du champs is defined by Γ(f, g) = 1 2 [A(f )g + f A(g) A(f g)]. Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 14 / 35

15 Analytic properties: Riesz transform Theorem (Lust-Piquard 99, Junge Z 15) (a) Let 2 p <. Then for every f Dom(A), c 1 p A 1/2 f p max{ Γ(f, f) 1/2 p, Γ(f, f ) 1/2 p } K p A 1/2 f p where c p = O(p 2 ) and K p = O(p 3/2 ). (b) Let 1 < p 2. Then for every f Dom(A), K 1 p A 1/2 f p inf { E(g g) 1/2 p + E(hh ) 1/2 p } C p A 1/2 f, δ(f)=g+h g G c p,h Gr p where K p = O(1/(p 1) 3/2 ) and C p = O(1/(p 1) 2 ). Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 15 / 35

16 Analytic properties: L p Poincaré inequalities Theorem (Junge Z 15) Let 2 p <. Then for every f Dom(A), f τ Q (f) p C p max{ Γ(f, f) 1/2 p, Γ(f, f ) 1/2 p }. Proofs follow from the Wick word decomposition theorem and the corresponding results in the matrix level. This idea was originally used by Biane ( 97) to deduce free hypercontractivity, and was later used by many authors. e.g. Kemp, Lee, Ricard, Junge, Palazuelos, Parcet, Perrin, et al. Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 16 / 35

17 CMAP and strong solidity A (finite) vna M has the weak* completely bounded approximation property (w*cbap) if there exists a net of normal, completely bounded, finite rank maps φ α : M M such that φ α cb C for all α and φ α id in the point weak* topology. The infimum of such constants C is called the Cowling Haagerup constant and is denoted by Λ cb (M). A vna with w*cbap is also said to be weakly amenable. If Λ cb (M) = 1, M is said to have the weak* completely contractive approximation property (w*ccap) or CMAP. Following Ozawa Popa, M is called strongly solid if the normalizer N M (P ) := {u U(M) : up u = P } of any diffuse amenable subalgebra P M generates an amenable vna. Here U(M) is the set of unitary operators in M. Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 17 / 35

18 Operator algebraic properties Theorem (Junge Z 15) Γ Q has w*ccap and is strongly solid provided max 1 i,j N q ij < 1. Some ideas in proof of CCAP: Find q such that max i,j q ij < q < 1. Let Q = q Q, where Q = ( q ij ) satisfies max i,j q ij < 1. Let X q i = l i + li and x i,j = X Q 1 n (f i e j ). Let π U : Γ Q (R N ) m,u Γ q(r m ) Γ Q 1m be a -homomorphism given by ( π U (X Q 1 i ) = m. X q m k i,k) x Then π U is trace preserving. Therefore, Γ Q is isomorphic to the von Neumann algebra generated by π U (X Q i ). k=1 Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 18 / 35

19 Some ideas in the proof Avsec ( 11): a net of finite rank maps ϕ α (A) : Γ q (R m ) Γ q (R m ), ϕ α (A) id in the point weak* topology, ϕ α (A) cb 1 + ε. Γ Q π U m,u Γ q(r m ) Γ Q 1m ψ α Γ Q π U ϕ α(a) id m,u Γ q(r m ) Γ Q 1m Strong solidity is more complicated and follows literally the same strategy of Houdayer Shlyakhtenko ( 11) which is an extension of Ozawa Popa ( 10). Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 19 / 35

20 Free transport: Background A general question: What s the relation between Γ Q (R N ) and L(F N ) = Γ 0 (R N ). A breakthrough of Guionnet Shlyakhtenko (2013(4)) develops a free transport theory and, together with Dabrowski s result on the existence of conjugate variables, proves that Γ q (R N ) = L(F N ) provided q is small enough. Suppose X = (X n ) n I is a sequence of algebraically free self-adjoint operators generating a tracial von Neumann algebra (M, τ). Let P denote the noncommutative polynomials in X n, n I. Voiculescu defined for each n the n-th free difference quotient n : P P P op by n (X k ) = δ n=k 1 1 n (AB) = n (A) B + A n (B), A, B P. Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 20 / 35

21 Free transport: Set-up Voiculescu ( 02) defined for each n N the n-th cyclic derivative D n : P P by D n (p) = BA, p=ax nb for p P a monomial and extend linearly to P. For P P, the sequence DP := (D n P ) n I is called the cyclic gradient of P. The n-th conjugate variable is ξ n L 2 M such that P, ξ n τ = n P, 1 1 τ τ op, P P. Clearly, ξ n = n(1 1), provided it exists. For a polynomial P = p monomial c pp P and for each R > 0, define P R = p c p R deg(p). Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 21 / 35

22 Free transport Let X = (X n ) n I be operators in a vna M with a faithful normal state ϕ. Recall that the joint law of X with respect to ϕ is a linear functional ϕ X defined on noncommutative polynomials by ϕ X (T i1 T id ) = ϕ(x i1 X id ), i I d. Let Z = (Z n ) n I be another sequence in a vna N with a faithful normal state ψ, and let ψ Z be the joint law of Z with respect to ψ. Observe that if ϕ X = ψ Z then W (X n : n I) = W (Z n : n I), since ϕ and ψ are faithful normal states. Definition (Guionnet Shlyakhtenko) Transport from ϕ X to ψ Z is a sequence Y = (Y n ) n I W (X n : n I) whose joint law with respect to ϕ is equal to ψ Z. That is, ϕ Y = ψ Z. Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 22 / 35

23 Free transport theorem of Guionnet Shlyakhtenko Theorem (Guionnet Shlyakhtenko 14) Let R > R > 4. Let X 1,..., X N (M, τ) be semicircular variables. Then there exists a universal constant C = C(R, R ) > 0 such that whenever W P (R+1) satisfies W R+1 < C, there is G P (R ) so that 1 If we set Y j = D j G, then Y 1,..., Y N P (R ) has law τ V, with V = 1 2 X 2 j + W ; 2 S j = H j (Y 1,..., Y n ) for some H P (R ) ; In particular, there are trace preserving isomorphisms C (τ V ) = C (X 1,..., X N ), W (τ V ) = L(F N ). Observation: Assume moreover that ξ i = ξ i = i (X i) belongs to P (R+1) for some R > 4. Voiculescu showed that V = 1 2 Σ( n j=1 X jξ j + ξ j X j ) satisfies ξ j = D j V and thus the theorem applies. Here Σ = N 1. Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 23 / 35

24 Isomorphism theorem Dabrowski showed that the conjugate variables ξ i, i = 1,..., N exist for q-gaussian variables (X q i ) provided q < q 0(N) small. Theorem (Guionnet Shlyakhtenko 14) Γ q (R N ) = Γ 0 (R N ) = L(F N ) and C (X q 1,..., Xq N ) = C (S 1,..., S N ) as long as q < q 0 (N). How about Γ Q (R N )? Theorem (Nelson Z 15a) Let Q = (q ij ) be a symmetric N N matrix with N {2, 3,...} and q ij ( 1, 1). Then Γ Q (R N ) = Γ 0 (R N ) = L(F N ) and C (X Q 1,..., XQ N ) = C (S 1,..., S N ) as long as max i,j q ij < q 0 (N). Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 24 / 35

25 Proof of isomorphism theorem: Some ideas With Guionnet Shlyakhtenko s transport theorem, we only need to construct suitable conjugate variables. Or, extend the argument of Dabrowski to the mixed q case. The derivation (Q) j (Q) j : C X Q 1,..., XQ N B(L2 (Γ Q (R N ))), (Q) j (X) = [X, r j ] := Xr j r j X. The derivation Ξ i Ξ i : F Q (R N ) F Q (R N ), Ξ i (e j1 e jn ) = q ij1 q ijn e j1 e jn. (NB: If q ij q, Ξ 1 = Ξ 2 =.) Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 25 / 35

26 Proof of isomorphism theorem: Some ideas If q is small enough, Ξ i is invertible and Ξ 1 i can be written as a noncommutative power series. This is based on a crucial estimate of Bożejko ( 98) (see Dykema Nica 93 for fixed q case): (P (n) ) 1 [ (1 q) k=1 1 + q k ] n. 1 q k Here ξ, η Q = δ n,m ξ, P (n) η 0 for ξ (R N ) n, η (R N ) m ; q = max 1 i,j N q ij. ξ j (Y 1,..., Y N ) := (Ξ 1 j ) (Y 1,..., Y N )#Y j m (1 τ Q 1) (1 (Q) j + (Q) j where (a b)#x = axb and m(a b) = ab. 1)[(Ξ 1 j ) (Y 1,..., Y N )], Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 26 / 35

27 Proof of isomorphism theorem: Some ideas Voiculescu ( 98) implies n (Q) ( ) = n ( )#Ξ n and Define ξ j, P τq = (Ξ 1 j V (Y 1,..., Y N ) = Σ ξ j := ξ j (X 1,..., X N ) = (Q) j ((Ξj 1 ) ). ( 1 2 ), (Q) j (P ) HS = 1 1, j (P ) τq τ op; Q ) N ξ i (Y 1,..., Y N )Y i + Y i ξ i (Y 1,..., Y N ). i=1 Guionnet Shlyakhtenko s theorem yields the isomorphism result. Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 27 / 35

28 The case of infinite variables Question (Voiculescu, others(?)): Are the methods of Guionnet Shlyakhtenko valid for an infinite number of variables? The difficulty: In the fixed q case, Dabrowski s theorem on the existence of conjugate variables requires q < q 0 (N) and q 0 (N) 0 as N. So free transport for infinite variables (assume it is valid) does not apply to Γ q (l 2 ). Note also that the structure array (q ij q) i,j N of Γ q (l 2 ) is not even bounded as an operator on l 2 unless q = 0. So (q ij q) i,j N is not small which is required in free transport. However, if q ij decays very fast, one may have hope. Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 28 / 35

29 Some infinite variable formalism Let P (R) denote l (N, P (R) ), the set of uniformly bounded sequences of elements of P (R), with norm Definition (P n ) n N R, := sup P n R. n Given W P (R) for R > sup n X n, we say that τ X is a free Gibbs state with quadratic potential perturbed by W (or a free Gibbs state with perturbation W ) if τ([x n + D n W ]P ) = τ τ op ( n P ) P P, n N. Note that V 0 = 1 2 N i=1 X2 i C X 1,..., X N does not converge in R-norm as N, we have modified the definition to refer only to the perturbation W P (R). Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 29 / 35

30 Free transport for infinite variables Theorem (Nelson-Z 15b) Let X = (X n ) n N be free semicircular variables generating the von Neumann algebra M = L(F ), with trace τ, and let R > S > 4. Suppose N is a von Neumann algebra with a faithful normal state ψ, and the joint law of Z = (Z n ) n N N with respect to ψ is a free Gibbs state with R+1 e log( S+1 ) perturbation W = W = P (R+1). If W R+1 2, then transport from τ X to ψ Z is given by Y = X + Dg P (S) for some g = g P (S). This transport satisfies Y X S, 0 as W R+1 0, and is invertible in the sense that H(Y ) = X for some H P (2.4). In particular, there are trace-preserving isomorphisms: C (Z n : n N) = C (X n : n N) and W (Z n : n N) = L(F ). Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 30 / 35

31 Isomorphism result Q i (p) := j 1 q ij p. π(q, n, R) := Theorem (Nelson-Z 15b) [(R(1 q)+1)q n( 1 2 )]2 (1 2q) 2 [(R(1 q)+1)q n( 1 2 )]2. Let R > 5. If the structure array Q for the mixed q-gaussian algebra Γ Q satisfies 0 < π(q, n, R) < 1 for all n N, and ( π(q, n, R) 1 π(q, n, R) < e log R ) ( 5 ), 4 R R + n N R sup n X Q n then Γ Q = L(F ) and C (X Q n : n N) = C (X n : n N), where {X n } n N is a free semicircular family. Example: If q ij = q i+j 1, then one can take q < and R = 6.7. Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 31 / 35

32 Some words about the proof The construction of free transport follows the same steps as Guionnet Shlyakhtenko along with some technical extensions and modification suitable to infinite variable setting. e.g. if R > S > max{1, sup n X n }, n ΣP R πr 1 R P R and n P S πs C (R, S) P R, n I n I where R πr is the projective tensor norm on P (R) (P (R) ) op : { η R πr := inf A i R B i R : η = } A i B i. i i Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 32 / 35

33 Some words about the proof The proof of the existence of conjugate variables (and potential) follows similar strategy used by Dabrowski and our previous work on finite variable case along with more careful analysis of certain estimation. For example, in the finite variable case, if q := max 1 i,j N q ij satisfies q 2 N < 1 then Ξ j HS(F Q ). In the infinite variable case, suppose Q n (2) < 1. Then Ξ n is a Hilbert Schmidt operator with Ξ n HS = (1 Q n (2)) 1/2. Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 33 / 35

34 Open problems So far certain small perturbations of free semicircular systems in Bożejko Speicher algebras have been understood. How about regimes far away from q = 0? Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 34 / 35

35 Thank you for your attention! Qiang Zeng (Northwestern University) Mixed q-gaussian and free transport 35 / 35