De Finetti theorems for a Boolean analogue of easy quantum groups 1 - PowerPoint PPT Presentation

De Finetti theorems for a Boolean analogue of easy quantum groups 1 / 27

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

Free and Boolean de Finetti theorems Free and Boolean de Finetti theorems : 1 Free de Finetti theorem for A s ( C. K¨ ostler 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

De Finetti theorems for free quantum groups ( M ,ϕ ) : v.N.alg and faithful normal state x n ∈ M s . a . ( n ∈ N ) ( x n ) n ∈ N is Invariant under iff S + free i.i.d. over tail (*) n O + (*) & centered semicircular n B + (*) & semicircular n H n (*) & even Symmetries Categories of partitions Distributions S + free i.i.d. over tail (*) NC n O + (*) & centered semicircular NC 2 n B + (*) & semicircular NC b n 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

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 j 1 ) f 2 ( x j 2 ) ⋯ f k ( x j k )] = E [ f 1 ( x j 1 )] E [ f 2 ( x j 2 )] ⋯ E [ f k ( x j k )] , whenever j 1 ≠ j 2 ≠ ⋯ ≠ j k and f 1 ,..., f k ∈ N ⟨ X ⟩ ○ . ( i . e . N − polynomials without constant terms ) 4 / 27

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 n U ij P = P , U ij P = P , ∑ ∑ j = 1 i = 1 U i 1 j U i 2 j = 0 , if i 1 ≠ i 2 , U ij 1 U ij 2 = 0 , if j 1 ≠ j 2 . Theorem (Liu, 2015) ( 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 σ w E tail ∶ M → M tail ∶= ⋂ ∞ n = 1 ev x ( P o ≥ n ) and ( x j ) j ∈ N is Boolean i.i.d. over tail. 5 / 27

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

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 } ∶ Horizontal concatenation p ⊗ q = { Q P } − { closed blocks } ∶ Vertical concatenation = pq { P ↷ } ∶ Upside-down turning p ∗ = 7 / 27

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

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

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 , ... ∈ NC x (∣ V ∣) . �ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ�ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ� ∣ 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 M¨ obius inversions: 10 / 27

Review on M¨ obius function Let ( Q , ≤ ) be a finite poset. The M¨ obius 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

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

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

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 )} =∶ 2 n D − 1 , we have ... ... ∈ D ( 0 , k ) � ⇒ ∈ D ( 0 , k − 2 ) . �ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ�ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ� �ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ�ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ� k − 2 k Consider the case k is odd, k ≠ 2 n D − 1. We have the following inequalities among partitions. ... ... ≤ ... ≤ ... �ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ�ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ� 1 2 n D − 1 1 k − 2 1 k k + 1 2 − n D By block-stable property, 1 k − 2 ⊗ ∈ D . 14 / 27

Classification D : blockwise category of interval partitions L D ∶= { k ∈ N ∣ 1 k ∈ D ( k )} l D ∶= sup { l ∈ N ∣ 2 l ∈ L D } . ⎧ ⎪ sup { m ∈ N ∣ 2 m − 1 ∈ L D } , ⎪ if L D contains some odd numbers, m D ∶= ⎨ ⎪ ∞ , ⎪ otherwise . ⎩ ⎧ ⎪ min { m ∈ N ∣ 2 m − 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

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 ∶= ... ∈ D ( k ) , �ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ�ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ� k ⎧ ⎪ j 1 = ⋯ = j k , n ⎪ p , u ij 1 ⋯ u ij k p = ⎨ ∑ ⎪ ⎪ 0 , otherwise, ⎩ i = 1 ⎧ ⎪ n ⎪ i 1 = ⋯ = i k , p , u i 1 j ⋯ u i k j p = ⎨ ∑ ⎪ ⎪ 0 , otherwise. ⎩ j = 1 16 / 27

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 X i 1 ⋯ X i k ⊗ pu i 1 j 1 ⋯ u i k j k p , j ∈ [ n ] k Ψ n ( X j 1 ⋯ X j k ) ∶= ∑ i ∈ [ n ] k Ψ n is a coaction, that is, ( Ψ n ⊗ id ) ○ Ψ n = ( id ⊗ ∆ ) ○ Ψ n . 17 / 27