De Finetti theorems for a Boolean analogue of easy quantum groups

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2 De Finetti theorems for a Boolean analogue of easy quantum groups Tomohiro Hayase Graduate School of Mathematical Sciences, the University of Tokyo March, 2016 Free Probability and the Large N limit, V The University of California, Berkeley 1 / 27

3 Free and Boolean de Finetti theorems Free and Boolean de Finetti theorems: 1 Free de Finetti theorem for A s (C. Köstler and R. Speicher, 2009) 2 Free de Finetti theorems for free quantum groups (T. Banica, S. Curran and R. Speicher, 2012) 3 Boolean de Finetti theorem for B s (W.Liu, 2015) Our result: Find general Boolean de Finetti theorem for a Boolean analogue of free quantum groups. Our strategy: Find a nice class of interval partitions and use BCS s framework. Liu himself proved Boolean de Finetti theorems for quantum semigroups by a different way. 2 / 27

4 De Finetti theorems for free quantum groups (M, ϕ) : v.n.alg and faithful normal state x n M s.a. (n N) Invariant under iff (x n ) n N is S n + free i.i.d. over tail (*) O n + (*) & centered semicircular B n + (*) & semicircular (*) & even H n Symmetries Categories of partitions Distributions S n + NC free i.i.d. over tail (*) O n + NC 2 (*) & centered semicircular B n + NC b (*) & semicircular H n NC h (*) & even Tannaka-Klein duality : A sequence of free quantum groups 1 1 (A x (n)) n N A category of noncrossing partitions NC x Cumulants-Moments formula 3 / 27

5 Review on conditional Boolean independence Definition η N M : a normal embedding of v.n. algebras w/ η(1 N ) /= 1 M, E M N : a normal conditional expecation w/ E η = id N. (x j M s.a. ) j J is Boolean independent w.r.t. E if E[f 1 (x j1 )f 2 (x j2 ) f k (x jk )] = E[f 1 (x j1 )]E[f 2 (x j2 )] E[f k (x jk )], whenever j 1 j 2 j k and f 1,..., f k N X. (i.e. N polynomials without constant terms) 4 / 27

6 Liu s Boolean de Finetti theorem Liu defined a quantum semigroup B s (n) as the universal unital C -algebra generated by projections P, U i,j (i, j = 1,..., n) and relations such that n j=1 Theorem (Liu, 2015) n U ij P = P, U ij P = P, i=1 U i1 ju i2 j = 0, if i 1 i 2, U ij1 U ij2 = 0, if j 1 j 2. (M, ϕ) : a v.n.algebra & a nondegenerate normal state. x j M s.a., j N with M = W (ev x (P o )) where P o = {f C (X j ) j N f (0) = 0} TFAE. 1 The joint distribution of (x j ) j N is invariant under the coaction of B s. 2 There exists a normal conditional expectation E tail M M tail = n=1 ev x(p o n) σw and (x j ) j N is Boolean i.i.d. over tail. 5 / 27

7 Our strategy Aim : Fill the missing piece in Boolean de Finetti theorem. Our strategy : Find a nice class of interval partitions and use BCS s framework. Difficulity: Bad-behavors of non-unital embeddings and non-faithful states 6 / 27

8 Review on category of partitions P(k, l): the set of all partitions of the disjoint union [k] [l], where [k] = {1, 2,..., k} for k N. Such a partition will be pictured as 1... k p = P 1... l where P is a diagram joining the elements in the same block of the partition. Categorical operations: p q = {PQ} Horizontal concatenation pq = { Q } {closed blocks} Vertical concatenation P p = {P } Upside-down turning 7 / 27

9 category of interval partitions NC = (NC(k, l)) k,l : the family of all noncrossing partitions NC x = {NC x (k, l)} k,l, NC x (k, l) NC(k, l) is a category of noncrossing partitions if 1 It is stable by categorical operations 2 NC x (0, 2) 3 NC x (1, 1) I (k) = {π P(k) interval partition}, I = (I (k) I (l)) k,l Definition (Category of interval partitions) I x = {I x (k, l)} k,l, I x (k, l) I (k, l) is a category of interval partitions if 1 It is stable by categorical operations 2 I x (0, 2) 8 / 27

10 Category of interval partitions Remark I x (k, l) = I x (k, 0) I x (0, l) I x (k) = I x (0, k) Example The followings are categories of interval partitions. 1 I 2 = ({π I (k) block size 2}) k 2 I b = ({π I (k) block size 2}) k 3 I h = ({π I (k) block size even}) k 9 / 27

11 Review on NC x To find the class of interval partitions suited to de Finetti, review on NC x. NC, NC 2, NC b, and NC h are block-stable, i.e. for any π NC x and V π,... V NC x ( V ). These four categories of noncrossing partitions are also closed under taking an interval in NC, i.e. ρ, σ NC x (k), π NC(k), ρ π σ π NC x (k). This condtition appears in Möbius inversions: 10 / 27

12 Review on Möbius function Let (Q, ) be a finite poset. The Möbius function µ Q {(π, σ) Q 2 π σ} C is defined by the following relations: for any π, σ Q with π σ, µ Q (π, ρ) = δ(π, σ), ρ Q π ρ σ µ Q (ρ, σ) = δ(π, σ), ρ Q π ρ σ where if π = σ then δ(π, σ) = 1, otherwise, δ(π, σ) = 0. Closed under taking an interval If R Q is closed under taking an interval in Q, µ R (π, σ) = µ Q (π, σ). 11 / 27

13 Blockwise condition We define a suitable class of interval partitions. Definition (Blockwise condition) Let D be a category of interval partition. D is said to be blockwise if 1 D is block-stable, 2 D is closed under taking an interval in I, i.e., ρ, σ D(k), π I (k), ρ π σ π D(k). Key condition If D is blockwise, µ D(k) (π, σ) = µ I (k) (π, σ). 12 / 27

14 Pairing By composition with the pair partition & the unit partition, it holds that k... NC x (0, k)... k 2 NC x (0, k 2). I x : a category of interval partitions Becasue the unit partition / I x (1, 1), in general, k... I x (0, k) /... k 2 I x (0, k 2). 13 / 27

15 Pairing in blockwise category of interval partition Lemma D : a blockwise category of interval partitions If k even & k > 2, or k odd& k > min{k 1 k D(k)} = 2n D 1, we have k... D(0, k)... k 2 D(0, k 2). Consider the case k is odd, k 2n D 1. We have the following inequalities among partitions nD 1... k+1 2 n D... 1 k k By block-stable property, 1 k 2 D. 14 / 27

16 Classification D: blockwise category of interval partitions L D = {k N 1 k D(k)} l D = sup{l N 2l L D }. sup{m N 2m 1 L D }, if L D contains some odd numbers, m D =, otherwise. min{m N 2m 1 L D }, if L D contains some odd numbers, n D =, otherwise. By lemma, we have 1 m D n D l D if n D. 2 l D m D + n D 1. And D is determined by l D, m D and n D. 15 / 27

17 A Boolean analogue of free quantum groups Definition D : a blockwise category of interval partitions. C(G D n ) := -algebra generated by p, u ij (1 i, j n) with p = p = p 2, u ij = u ij and the following relations: for any k with 1 k =... n i=1 n j=1 D(k), k p, j 1 = = j k, u ij1 u ijk p = 0, otherwise, p, i 1 = = i k, u i1 j u ik jp = 0, otherwise. 16 / 27

18 Notations on C(G D n ) Set a *-hom C(G D n ) C(G D n ) C(G D n ) by n (u ij ) = u ik u kj, k=1 (p) = p p. is a coproduct: (id ) = ( id). Set P o = the *-algebra of all nonunital polynomials in noncommutative countably infinite many variables (X j ) j N. We can define a linear map Ψ n P o P o C(G D n ) as the extension of Ψ n (X j1 X jk ) = i [n] k X i1 X ik pu i1 j 1 u ik j k p, j [n] k Ψ n is a coaction, that is, (Ψ n id) Ψ n = (id ) Ψ n. 17 / 27

19 Fixed point algebra Denote by P Ψn the fixed point algebra: P Ψn = {f P o f = f p}. We have P Ψn = Span{X π P o π D(k), k N}, where X π = j [n] k X j1 X jk. By this representation of P Ψn, there is a π ker j functional h on the subspace S D n satisfying (id h) = (h id) = h. Define a linear map E n P o P Ψn by E n = (id h) Ψ n. 18 / 27

20 Invariance (M, ϕ) : a v.n.algebra & a nondegenerate normal state. Definition (x j M s.a. ) j N is said to have G D -invariant joint distribution if (ϕ ev x id) Ψ n = ϕ ev x p. 19 / 27

21 Main Theorem Theorem (M, ϕ) : a v.n.algebra & a nondegenerate normal state. x j M s.a., j N with M = W (ev x (P o )) For any blockwise category of interval partitions D, TFAE. 1 The joint distribution of (x j ) j N is G D -invariant. 2 (x j ) j N is Boolean i.i.d. over tail, & for any k with 1 k D(k), K E tail k [x 1 b 1, x 1 b 2,..., x 1 ] = 0, b 1,, b k M tail {1}. In particular, Symmetries Categories of partitions Distributions Gn I I Boolean i.i.d. over tail (*) G I 2 n I 2 (*) & centered Bernoulli G I b n I b (*) & Bernoulli G I h n I h (*) & even 20 / 27

22 Strategy Assume the joint distribution of (x j ) j N is G D -invariant. Since G D -invariance implies B s -invariance, there exist a normal c.e. E tail M M tail given by E tail = e tail ( )e tail. ISTS for any b 0,..., b k M tail {1}, j [n] k, and k N, E tail [x j1 b 1 x j2 b 2 b k 1 x jk ] = Main strategy of the proof: σ D(k) σ ker j K E tail σ [x 1 b 1, x 1 b 2,..., x 1 ]. 1 Examine E tail can be approximated by E n = (id h)ψ n 2 Use Weingarten estimate If D is blockwise then µ D(k) = µ I (k). By using this, h(pu i1 j 1 u ik j k p) = π,σ D(k) π ker i σ ker j 3 Apply moments-cumulants formula. 1 n π [µ I (k)(π, σ) + O( 1 )] (as n ) n 21 / 27

23 Difficulty 1 : Coaction is non-multiplicative Since the coaction Ψ n is non-multiplicative : there exist b 1,..., b k 1 P Ψn Ψ n (f (X )g(x )) /= Ψ n (f (X ))Λ n (g(x )), with Ψ n [X j1 b 1 X j2 b 2 b k 1 X jk ] /= i [n] k X i1 b 1 X i2 b 2 b k 1 X ik pu i1 j 1... u ik j k p, So it is difficult to approximate E tail [x j1 b 1 b k 1 x jk ] by E n [X j1 b 1 X j2 b 2 b k 1 X jk ]. idea: By block-stable condition, and since E tail satisfies E tail = e tail ( )e tail, the following holds; Assume for any j [n] k and k N, E tail [x j1 x jk ] = σ D(k) σ ker j K E tail σ [x 1,..., x 1 ]. Then for any b 0,..., b k M tail {1}, j [n] k, and k N, E tail [x j1 b 1 x j2 b 2 b k 1 x jk ] = σ D(k) σ ker j K E tail σ [x 1 b 1, x 1 b 2,..., x 1 ]. 22 / 27

24 Difficulty 2 Difficulty 2 : As the state ϕ is non-faithful, we cannot define E n on M and cannot approximate E tail by E n. idea: e n = the orthogonal projection onto ev x (P Ψn )Ω ϕ. If we prove L 2 -lim n ev x (E n [X j1 X j2 X jk ])Ω ϕ = E tail [x j1 x j2 x jk ]Ω ϕ (j [n] k, k N) Then s-lim e n = e tail and hence s lim n ev x (E n [X j1 X j2 X jk ])e n = E tail [x j1 x j2 x jk ](j [n] k, k N) 23 / 27

25 Difficulties and key ideas Difficulty1 : As the state is non-faithful, we cannot define E n on M and cannot approximate E tail by E n. Idea : ISTS lim E tail[x j1 x j2 x jk ] = L 2 lim ev x E n [X j1 X j2 X jk ]. n n Difficulty2 : Coactions are non-multiplicative. Hence it is difficult to estimate E tail [x j1 b 1 x j2 b 2 b k 1 x jk ] (b 0,..., b k M tail {1}). Idea : ISTS E tail [x j1 x jk ] = K E tail σ [x 1,..., x 1 ]. σ D(k) σ ker j 24 / 27

26 Main Theorem Theorem (M, ϕ) : a v.n.algebra & a nondegenerate normal state. x j M s.a., j N with M = W (ev x (P o )) For any blockwise category of interval partitions D, TFAE. 1 The joint distribution of (x j ) j N is G D -invariant. 2 (x j ) j N is Boolean i.i.d. over tail, & for any k with 1 k D(k), K E tail k [x 1 b 1, x 1 b 2,..., x 1 ] = 0, b 1,, b k M tail {1}. In particular, Symmetries Categories of partitions Distributions Gn I I Boolean i.i.d. over tail (*) G I 2 n I 2 (*) & centered Bernoulli G I b n I b (*) & Bernoulli G I h n I h (*) & even 25 / 27

27 C -closure Free case: By Tannaka-Klein duality for compact quantum groups, Free quantum groups A x NC x. A x (n) = C univ(u = (u ij ) u t u = t uu = 1)/relations implied by NC x C X j j N ΨAx n = Span{X π P o π NC x }. Boolean case: C univ (p, u = (u ij) relations implied by D) can be ill-defined. Liu: B o (n) = C univ (p, u = (u ij) p = p = p 2, u t up = t uu = p, u 1) It is not clear P ΨBo n? = Span{X π P o π I 2 }. Hence h and E n can be changed, it is not obvious that our strategy works well for B o (n). 26 / 27

28 Summary Aim : Prove general Boolean de Finetti theorem. Our strategy : Find a nice class of interval partitions and use BCS s framework. Key condition: D is blockwise i.e. block-stable and closed under taking an interval in I. Second condition implies µ D(k) (π, σ) = µ I (k) (π, σ), π, σ D(k). Difficulty1 : As the state ϕ is non-faithful, it is difficult to define E n on M and approximate E tail by E n. Idea : ISTS E tail [x j1 x j2 x jk ] = L 2 lim n ev x E n [X j1 X j2 X jk ]. Difficulty2 : Coactions are non-multiplicative. Hence it is difficult to estimate E tail [x j1 b 1 x j2 b 2 b k 1 x jk ] (b 0,..., b k M tail {1}). Idea : By block-stable condition, ISTS E tail [x j1 x jk ] = σ D(k) σ ker j K E tail σ [x 1,..., x 1 ]. 27 / 27