Quantum Bernstein's Theorem and the Hyperoctahedral Quantum Group
aa r X i v : . [ m a t h . OA ] J un QUANTUM BERNSTEIN’S THEOREM AND THE HYPEROCTAHEDRALQUANTUM GROUP
PAWE L J ´OZIAK AND KAMIL SZPOJANKOWSKI
Abstract.
We study an extension of Bernstein’s theorem to the setting of quantum groups. Fora d -tuple of free, identically distributed random variables we consider a problem of preservationof freeness under the action of a quantum subset of the free orthogonal quantum group. Fora subset not contained in the hyperoctahedral quantum group we prove that preservation offreeness characterizes Wigner’s semicircle law. We show that freeness is always preserved if thequantum subset is contained in the hyperoctahedral quantum group. We provide examples ofquantum subsets which show that our result is an extension of results known in the literature. Introduction
The well-known classical Bernstein’s theorem states that given a d -dimensional random vector X = ( X , . . . , X d ) with independent coordinates and a generic orthogonal transformation (withthe exception of those which are equivalent to signed permutations) A ∈ O d , if the coordinatesof the random vector AX are still independent, then the random vector necessarily consists ofidentically distributed Gaussian random variables.This result forms one of the cornerstones of modern theory of random matrices. Indeed, if oneasks for an ensemble of random matrices, with independent entries on the upper triangular part,to be invariant under a change of basis (unitary in the complex case and orthogonal in the realcase), then one is forced to deal with Gaussian ensembles, due to Bernstein’s theorem. This can beseen as a no-go result: there is no generalization of GOE to non-Gaussian orthogonal ensembles.A result similar in nature to Bernstein’s theorem was obtained by Nica: he showed in [Nic96]that if a vector consists of free random variables that remain free after application of a genericrotation, then elements of this vector form necessarily a semicircular system (free probabilityanalogue of Gaussian random variables).The aim of this note is to push this study further. We ask whether a similar result can beobtained if one replaces orthogonal transformations O d by free quantum orthogonal transforma-tions O + d of Wang and van Daele [VDW96], and what is the right analogue of generic orthog-onal transformations in this context, i.e. which transformations do not characterize Wigner’slaw. This result can be seen as a sequel of series of recent works on applications of quantumgroup symmetry in the theory of free probability, originating from K¨ostler and Speicher’s freede Finetti theorem [KS09] (see also a survey [Spe14]) and later developed by Curran and others[BBC11, BCS11, BCS12, Cur09, Cur10, Cur11, CS11].Let O ( O + d ) be the Hopf ∗ -algebra of a free orthogonal quantum group O + d and let X ⊂ O + d beits subset, i.e. a ∗ -epimorphism β : O ( O + d ) → B = O ( X ) for some ∗ -algebra B . Let ( A , ϕ ) be anon-commutative ∗ -probability space and consider a family of random variables X , . . . , X d ∈ A .We will study the quantum family of rotated random variables by the transformations from X : therandom variables Y j = P di =1 X i ⊗ ˙ u ij ∈ A ⊗ O ( X ), where u = ( u ij ) ≤ i,j ≤ d is the fundamentalcorepresentation of O ( O + d ) and ˙ u ij = β ( u ij ). Let us also denote by H + d the hyperoctahedralquantum group [BBC07]. The main result of this note states the following. Theorem (Quantum Bernstein’s theorem) . Assume that X , . . . , X d are free and identically dis-tributed. If Y = P di =1 X i ⊗ ˙ u i , . . . , Y d = P di =1 X i ⊗ ˙ u id are free with amalgamation over O ( X ) Mathematics Subject Classification.
Primary: 46L54, 20G42, Secondary: 62E10, 46L89.
Key words and phrases.
Bernstein’s theorem, Compact quantum groups, Freeness, Free Cumulants, Wigner’slaw. and X H + d , then X , . . . , X d form a semicircular system. Conversely, given a d -tuple of freerandom variables X , . . . , X d ∈ ( A , ϕ ) , the d -tuple Y j = P di =1 X i ⊗ ˙ u ij ∈ A ⊗ O ( H + d ) is free withamalgamation over O ( H + d ) . Let us stress that, unlike the aforementioned articles on quantum groups as the sources ofdistributional symmetries in free probability, we do not assume invariance of the joint distribution.We only assume that the freeness property is preserved, the joint distribution may, a priori,change. The plan of the manuscript is as follows: we gather some preliminaries in Section 1:we recall relevant notions from the theory of quantum groups in Section 1.1 and from operator-valued free probability theory in Section 1.2; in particular we recall the notion of operator-valuedfree cumulants, one of the main tools which we use in the proof of the main result. Here is thepoint where we were not able to adopt in a straightforward way the proof of non-commutativeBernstein’s theorem - there is no notion of multidimensional R -transform in the operator-valuedsetting, for more details see [Spe98]. We are forced to use operator-valued free cumulants instead.Section 2 is the core of the paper: we start with the description of the change of coordinates in freecumulants under the action of the free orthogonal quantum group. Next we prove the first partof the main result stated above. The aim of Section 3 is to discuss optimality (Section 3.1) andnon-triviality (Section 3.2) of our quantum Bernstein’s theorem. Namely, we show that a randomvector consisting of free entries remains free after application of rotations from the hyperoctahedralquantum group, whatever the starting marginal distributions were. This proves second part ofthe Theorem above. If any quantum subset X ⊂ O + d contained classical generic rotation , ourresult would be reducible to Nica’s non-commutative Bernstein’s Theorem. In Section 3.2 weprovide examples of quantum subsets X ⊂ O + d satisfying assumptions of our quantum Bernstein’sTheorem, yet having not enough points for Nica’s result to be applicable.1. Preliminaries
Free orthogonal and hyperoctahedral quantum groups.
The theory of compact quan-tum groups in operator-algebraic setting was initiated by Woronowicz [Wor87] (see [Wor98, Tim08]for more details), we briefly introduce key concepts of this theory. A unital C ∗ -algebra A endowedwith a ∗ -homomorphism ∆ : A → A ⊗ A (the minimal tensor product) satisfying the coassociativitycondition: (∆ ⊗ id)∆ = (id ⊗ ∆)∆ is called a Woronowicz algebra , if the cancellation laws holds:( ⊗ A ) · ∆( A ) = A ⊗ A = ( A ⊗ ) · ∆( A ) . The Woronowicz algebra A can be always endowed with a unique state h ∈ A ∗ , called the Haarstate , which is left and right invariant:(id ⊗ h )∆ = ( h ⊗ id)∆ = h ( · ) . Such an algebra correspond to a compact quantum group G via abstract extension of the Gelfand-Naimark duality: A = C ( G ), the algebra of continuous functions on G . This algebra contains aunique dense Hopf ∗ -subalgebra O ( G ) (i.e. ∆ ↾ O ( G ) : O ( G ) → O ( G ) ⊗ alg O ( G )); it is spanned bymatrix coefficients of unitary representations of G . This Hopf ∗ -algebra can have many different C ∗ -completions: the norm induced by GNS construction for h , the one coming from A and theuniversal C ∗ -norm need not coincide. However, in our considerations only the Hopf-algebraicstructure of O ( G ), and existence of a faithful Hilbert space representation of O ( G ), are relevant.We call G a compact matrix quantum group if C ( G ) can be given a fundamental corepre-sentation u ∈ M n ( C ( G )) = B ( C n ) ⊗ C ( G ). Denoting u ij = ( h e i | · | e j i ⊗ id) u for a fixed ONB( e i ) ≤ i ≤ n ⊂ C n (with the standard inner product), u is called a fundamental corepresentation if:∆( u ij ) = n X k =1 u ik ⊗ u kj i, j ∈ { , . . . n } and h{ u ij : 1 ≤ i, j ≤ n }i = O ( G )where h X i denotes the ∗ -algebra generated by elements of X ⊆ C ( G ). Then the Hopf-algebraicstructure on O ( G ) is uniquely determined by the prescription S ( u ij ) = u ∗ ji and ε ( u ij ) = δ ij . UANTUM BERNSTEIN’S THEOREM AND THE HYPEROCTAHEDRAL QUANTUM GROUP 3
Sticking to the Gelfand-Najmark picture, a homomorphism H → G is, by definition, the trans-pose of a ∗ -homomorphism π : O ( G ) → O ( H ) which intertwine the coproducts: ( π ⊗ π ) ◦ ∆ G =∆ H ◦ π . If moreover π is a surjection, then we say that H is a subgroup of G and write H ⊂ G .In this article we will be mainly interested in three examples of compact matrix quantumgroups: the free orthogonal quantum group O + d , the hyperoctahedral quantum group H + d and thequantum group of symmetries of a cube O − ( d ). Definition 1.1 ([VDW96]) . Consider the universal C ∗ -algebra C u ( O + d ) generated by d genera-tors u ij , 1 ≤ i, j ≤ d subject to the relations:(1) all generators are self-adjoint u ij = u ∗ ij ;(2) the matrix u = ( u ij ) ≤ i,j ≤ d is orthogonal, i.e. u ⊤ u = uu ⊤ = ∈ M d ( C u ( O + d )).The C ∗ -algebra C u ( O + d ) is an algebra of continuous functions on a compact quantum group O + d ,where the group-structure on O + d is given the fundamental corepresentation u . The quantumgroup O + d is called the free orthogonal quantum group .For later reference, let us unpack condition (2):(1.1) d X i =1 u ij = = d X j =1 u ij and(1.2) d X i =1 u ij u ij ′ = 0 = d X j =1 u ij u i ′ j for all non-quantified indices i = i ′ or j = j ′ . Definition 1.2 ([BBC07]) . Consider the universal C ∗ -algebra C u ( O − ( d )) generated by d gen-erators u ij , 1 ≤ i, j ≤ d subject to the relations:(1) all generators are self-adjoint u ij = u ∗ ij ;(2) the matrix u = ( u ij ) ≤ i,j ≤ d is orthogonal, i.e. u ⊤ u = uu ⊤ = ∈ M d ( C u ( O − ( d )));(3) u ij u kl = u kl u ij unless i = k or j = l (that is, unless these element lie in the same columnor row in the matrix u , they commute);(4) u ij u ij ′ = − u ij ′ u ij if j = j ′ , and u ij u i ′ j = − u i ′ j u ij if i = i ′ (that is, different elements inthe same row/column anticommute).The C ∗ -algebra C u ( O − ( d )) is an algebra of continuous functions on a compact quantum group O − ( d ), where the group-structure on O − ( d ) is given by the fundamental corepresentation u . Definition 1.3 ([BBC07]) . Consider the universal C ∗ -algebra C u ( H + d ) generated by d generators u ij , 1 ≤ i, j ≤ d subject to the relations:(1) all generators are self-adjoint u ij = u ∗ ij ;(2) the matrix u = ( u ij ) ≤ i,j ≤ d is orthogonal, i.e. u ⊤ u = uu ⊤ = ∈ M d ( C u ( H + d ));(3) for all i, j one has u ij = u ij (or, equivalently, u ij = u ij , or, equivalently, σ ( u ij ) ⊂ {± , } );(3’) u ij u ij ′ = 0 = u ji u j ′ i for all i, j = j ′ .The C ∗ -algebra C u ( H + d ) is an algebra of continuous functions on a compact quantum group H + d ,where the group-structure on H + d is given by the fundamental corepresentation u . The quantumgroup H + d is called the hyperoctahedral quantum group . Remark . Note that relations (3) and (3 ′ ) are equivalent and it is redundant to include bothof them in the definition. On the other hand, in some computations one or the other will be moreuseful, we put them both for convinience of the reader. The equivalence of (3) and (3 ′ ) was statedin [BBCC11, Proposition 11.4(3)], and the sketch of the passage between the conditions is asfollows: computing ( P i u ij ) , using (1.1), (3 ′ ) and positivity of u ij ( − u ij ) u ij one arrives at (3).Conversly, sum of projections is a projection if and only if they are mutually orthogonal, hence(1.1) and (3) implies u ij u ij ′ = 0. Then the C ∗ -identity, self-adjointness and the standard formula PAWE L J ´OZIAK AND KAMIL SZPOJANKOWSKI for spectral radius (which is equal to the norm for a self-adjoint element) yield the equality of theleft-hand side of (3 ′ ), the right-hand side follows from applying antipode S . Remark . Observe that H + d , O − ( d ) ⊂ O + d . Let us denote the ideals (in O ( O + d )) generated bythe relations defining H + d and O − ( d ) by I H + d and I O − ( d ) , respectively, observe that I O − ( d ) ⊂ I H + d .Indeed, I H + d = ≪ { u ij u ij ′ : i, j = j ′ }∪{ u ij u i ′ j : i = i ′ , j } ≫ , whereas I O − ( d ) = ≪ { u ij u ij ′ + u ij ′ u ij : i, j = j ′ }∪{ u ij u i ′ j + u i ′ j u ij : i = i ′ , j }∪{ u ij u i ′ j ′ − u i ′ j ′ u ij : i = i ′ , j = j ′ } ≫ , where ≪ X ≫ denotesthe ideal generated by X . Thus the canonical map O ( O + d ) ∋ u ij u ij ∈ O ( H + d ) factor through O ( O + d ) ∋ u ij u ij ∈ O ( O − ( d )). In other words, we have H + d ⊂ O − ( d ) ⊂ O + d canonically.1.2. Non-commutative probability.
Here we recall basic definitions and facts concerning freeprobability and its operator valued extension. For more details we refer to [NS06, Spe98, MS17].
Definition 1.6.
A non-commutative probability space consists of a pair ( A , ϕ ) where A is a unital ∗ -algebra and ϕ : A → C is linear functional such that ϕ ( ) = 1 and ϕ ( aa ∗ ) ≥ a ∈ A .In this paper we will deal only with compactly supported measures. Such measures are uniquelydetermined by their moment sequences and we will identify the distribution of a random variablevia moments. Definition 1.7.
For a self-adjoint random variable a ∈ A the distribution of a is the uniqueprobability measure such that ϕ ( a n ) = Z t n dµ ( t ) , for all n = 1 , , . . . .In the non-commutative setting it is possible to define new notions of independence. The mostprominent non-commutative independence is freeness defined by Voiculescu in [Voi86]. Definition 1.8.
Consider a NCPS ( A , ϕ ) and a family of unital subalgebras ( A i ) i ∈ I . Thesubalgebras ( A i ) i ∈ I are free if ϕ ( a · · · a n ) = 0 whenever a i ∈ A j i , j = j = . . . = j n and ϕ ( a i ) = 0 for all i = 1 , . . . , n and n = 1 , , . . . . Similarly, self-adjoint random variables a, b ∈ A arefree (freely independent) when subalgebras generated by ( a, ) and ( b, ) are freely independent.It turns out that for free random variables an analogue of Central Limit Theorem holds, thatis if one takes a sequence ( a n ) n ≥ of identically distributed, free random variables with mean zeroand variance one, then the distribution of the sequence a + . . . + a n √ n tends to a universal limit, which is Wigner’s semicircle law which has the density12 π p − x [ − , ( x ) . Free random variables can be succinctly studied in terms of the so called free cumulants.Let χ = { B , B , . . . } be a partition of the set of numbers { , . . . , k } . A partition χ is a crossingpartition if there exist distinct blocks B r , B s ∈ χ and numbers i , i ∈ B r , j , j ∈ B s such that i < j < i < j . Otherwise χ is called a non-crossing partition. The set of all non-crossingpartitions of { , . . . , k } is denoted by N C ( k ). Definition 1.9.
For any k = 1 , , . . . , (joint) cumulants of order k of non-commutative randomvariables a , . . . , a n are defined recursively as k -linear maps κ k : A k → C through equation ϕ ( a · . . . · a m ) = X π ∈ NC ( m ) Y B ∈ π κ | B | ( a i , i ∈ B )with | B | denoting the size of the block B . One can write the above as ϕ ( a · · · a n ) = X π ∈ NC ( n ) κ π ( a , . . . , a n )Cumulants of single variable a , are defined in the same manner as above, one takes a = . . . = a n = a . We denote κ n ( a ) = κ n ( a, . . . , a ). UANTUM BERNSTEIN’S THEOREM AND THE HYPEROCTAHEDRAL QUANTUM GROUP 5
With free cumulants in hand one can easily characterize the semicircle distribution; for detailswe refer to [NS06, Lecture 11].
Proposition 1.10.
A random variable a has the semicircular distribution with mean µ and vari-ance σ if and only if κ ( a ) = µ, κ ( a ) = σ and κ n ( a ) = 0 for all n ≥ . Next we discuss briefly extension of the concept of non-commutative probability space - socalled operator valued non-commutative probability space. Observe that when one takes B = C ,then we are in the scalar valued framework, discussed above. Definition 1.11. A B -valued non-commutative probability space (NCPS) consists of a triple( A , B , E ), where B ⊂ A and E is a conditional expectation. Definition 1.12.
For a ∗ -algebra A and its ∗ -subalgebra B a linear map E : A → B is called aconditional expectation if E ( b ) = b ∀ b ∈ B and E ( b ab ) = b E ( a ) b ∀ a ∈ A and ∀ b , b ∈ B Definition 1.13.
Let x , . . . , x d ∈ A , then the joint distribution of x , . . . , x d is given by all jointmoments of the form E ( y b . . . b n − y n ) , where y i ∈ { x , . . . , x d } and b i ∈ B , i = 1 , . . . , n and n ≥ Definition 1.14.
Consider a B -valued NCPS ( A , B , E ) and a family of subalgebras ( A i ) i ∈ I where B ⊂ A i for all i ∈ I . The subalgebras ( A i ) i ∈ I are free with amalgamation over B if E ( a · · · a n ) = 0whenever a i ∈ A j i , j = j = . . . = j n and E ( a i ) = 0 for all i = 1 , . . . , n and n = 1 , , . . . . Definition 1.15.
For an operator valued NCPS we define the corresponding B -valued cumulants( κ B n ) n ≥ via the moment-cumulant formula E ( a · · · a n ) = X π ∈ NC ( n ) κ B π ( a , . . . , a n )where cumulants are nested inside each other according to the nesting of blocks of π . Remark . For our purposes it is important that one can write an explicit formula for cumulantsin terms of moments κ B n ( a , . . . , a n ) = X π ∈ NC ( n ) E π ( a , . . . , a n ) µ ( π, n ) , (1.3)where E π is a multiplicative functional defined on the lattice N C ( n ) and again moments are nestedaccording to the nesting of blocks of π (for the explanation of nesting, see [MS17, p. 240]). By µ ( · , · ) we mean the M¨obius function on N C ( n ) and n is the maximal partition in N C ( n ) withrespect to the reversed refinement order.The relevance of operator valued cumulants stems from the following result proved in [Spe98]. Theorem 1.17.
Let ( A , B , E ) be a B -valued probability space and ( x i ) i ∈ I a family of randomvariables in A . Then the family ( x i ) i ∈ I is free with amalgamation over B if and only if κ B n ( y b , . . . , y n b n ) = 0 for every n ≥ , all b , . . . , b n ∈ B and every non-constant choice of y , . . . , y n ∈ { x i : i ∈ I } PAWE L J ´OZIAK AND KAMIL SZPOJANKOWSKI Quantum orthogonal transformations applied to random vectors
Assume X ⊂ O + d is a closed quantum subset of quantum orthogonal group, i.e. there is asurjective ∗ -homomorphism β : O ( O + d ) ։ O ( X ) (in particular, O ( X ) is generated, as an algebra,by elements β ( u ij ), which we denote later by ˙ u ij ).Let ( A , ϕ ) be an NCPS. Then on A ⊗ O ( X ) one can define a conditional expectation given by E = ϕ ⊗ id, and we obtain an operator valued NCPS ( A ⊗ O ( X ) , O ( X ) , E ). Remark . (Moments and cumulants of rotated variables) Take random variables X , . . . , X d ∈A , and define Y j = P di =1 X i ⊗ ˙ u ij ∈ A ⊗ O ( X ). Then by the definition of E and the fact that A and O ( X ), seen as subalgebras in A ⊗ O ( X ), commute, we have E ( Y j · · · Y j n ) = d X i ,...,i n =1 ϕ ( X i , . . . , X i n ) ˙ u i j · · · ˙ u i n j n . From the above we get that E π ( Y j · · · Y j n ) = d X i ,...,i n =1 ϕ π ( X i , . . . , X i n ) ˙ u i j · · · ˙ u i n j n , which in turn, by formula (1.3) (and simple change of order of summation) implies κ O ( X ) n ( Y j , . . . , Y j n ) = d X i ,...,i n =1 κ n ( X i , . . . , X i n ) ˙ u i j · · · ˙ u i n j n , (2.1)where κ O ( X ) n denote the operator-valued cumulants corresponding to the conditional expectation E and κ n are scalar valued cumulants corresponding to ϕ . Theorem 2.2.
Assume that X , X , . . . , X d are free and Y = P di =1 X i ⊗ ˙ u i , Y = P di =1 X i ⊗ ˙ u i , . . . , Y d = P di =1 X i ⊗ ˙ u id are free with amalgamation over O ( X ) . Assume moreover X i areidentically distributed. Then X , X , . . . , X d are semicircular random variables, unless X ⊂ H + d (i.e. the homomorphism β factors as O ( O + d ) → O ( H + d ) → O ( X ) ).Remark . We expect that the assumption of identical distribution of X i could be dropped, asin the classical Bernstein’s theorem. However, with the proof we found, it is essential. Proof.
The strategy of the proof is the following: freeness of Y , Y , . . . , Y d translates into relationsabout ˙ u ij ’s with coefficients κ n ( X i ) by means of Theorem 1.17. We show that, if these coefficientsdo not vanish for n ≥
3, then ˙ u ij = ˙ u ij , the defining relation for the hyperoctahedral quantumgroup, which violates the assumption X H + d .We start with even cumulants: assume n ≥ j = j ′ , by freeness of Y j and Y j ′ ,Theorem 1.17 yields: κ O ( X ) n ( Y j ′ , Y j ′ , Y j , . . . , Y j ) = 0where Y j ′ appears only in first two spots. By (2.1), this is equivalent to d X i ,...,i n =1 κ n ( X i , . . . , X i n ) ˙ u i j ′ ˙ u i j ′ ˙ u i j · · · ˙ u i n j = 0 , which simplifies, thanks to freeness of X , X , . . . , X d , to the formula d X i =1 κ n ( X i ) ˙ u ij ′ ˙ u n − ij = 0 . This formula is valid for any pair of indices j = j ′ . Sum them all (over j ′ = j ) to obtain (with theaid of (1.1)):(2.2) d X i =1 κ n ( X i )( X j ′ = j ˙ u ij ′ ) ˙ u n − ij = d X i =1 κ n ( X i )( − ˙ u ij ) ˙ u n − ij = 0 . UANTUM BERNSTEIN’S THEOREM AND THE HYPEROCTAHEDRAL QUANTUM GROUP 7
Now as κ n ( X i ) = κ n ( X ), either κ n ( X i ) = 0 or, by dividing (2.2) by κ n ( X ) and rearrangingthe terms, we have that: d X i =1 ˙ u n/ − ij ( − ˙ u ij ) ˙ u n/ − ij = 0The above formula is valid for any j . But as it is a sum of positive operators, this can happenonly if each of the summands is a zero operator, thus for all i, j we have that˙ u nij = ˙ u n +2 ij which, by spectral calculus, implies that σ ( ˙ u i,j ) ⊂ {± , } , and consequently ˙ u ij = ˙ u ij , as desired.We now proceed with the case n ≥ j = j ′ , by freeness of Y j and Y j ′ , Theorem 1.17 yields: κ O ( X ) n ( Y j , Y j ′ , Y j ′ , Y j , Y j , . . . , Y j n − ) = 0where Y j appears on the first spot, Y j appears only on the second and third spots, and each Y j , . . . , Y j n − appears exactly twice on neighboring spots. By (2.1), this is equivalent to d X i ,...,i n =1 κ n ( X i , . . . , X i n ) ˙ u i j ˙ u i j ′ ˙ u i j ′ · · · ˙ u i n j n − = 0 , which simplifies, thanks to freeness of X , X , . . . , X d , to d X i =1 κ n ( X i ) ˙ u ij ˙ u ij ′ ˙ u ij · · · ˙ u ij n − = 0 . The above relation is valid for any pair of indices j = j ′ and any choice of j , . . . , j n − . Aftersumming over all j , and then over all j , ..., and then over all j n − , and using (1.1), we obtain:(2.3) d X i =1 κ n ( X i ) ˙ u ij ˙ u ij ′ = 0 . The above is valid for all pairs j = j ′ . We can now use this relation in two different ways. Tosimplify things, we use the assumption κ n ( X i ) = κ n ( X ) = 0 and divide the relations by thisscalar. Firstly, summing over all j ′ = j , one obtains (similarly as in the case of n even): d X i =1 ˙ u ij ( − ˙ u ij ) = 0 , or, equivalently,(2.4) d X i =1 ˙ u ij = d X i =1 ˙ u ij . on the other hand, computing X ∗ X where X denotes the left hand side of (2.3), we arrive at: X i,i ′ ˙ u ij ′ ˙ u ij ˙ u i ′ j ˙ u i ′ j ′ = 0this relation is valid for all pairs j = j ′ . We divide the above sum into sum over i = i ′ and overall pairs ( i, i ′ ) such that i = i ′ . Then sum over all j = j ′ (with j ′ fixed) and change the order ofsummation to obtain: X i ˙ u ij ′ ( X j = j ′ ˙ u ij ) ˙ u ij ′ + X i,i ′ : i = i ′ ˙ u ij ′ ( X j = j ′ ˙ u ij ˙ u i ′ j ) ˙ u i ′ j ′ = 0 PAWE L J ´OZIAK AND KAMIL SZPOJANKOWSKI where the middle sum of the first term is nothing but − ˙ u ij ′ thanks to (1.1), and the sum in themiddle of the second term is nothing but − ˙ u ij ′ ˙ u i ′ j ′ thanks to (1.2). This can be rewritten as: X i ˙ u ij ′ = X i,i ′ ˙ u ij ′ ˙ u i ′ j ′ = X i ˙ u ij ′ ! where we moved the order-six terms from the first sum to the right hand side. Now we use (2.4)to change the right hand side. This yields: X i ˙ u ij ′ = ( X i ˙ u ij ′ ) = X i ˙ u ij ′ + X i,i ′ : i = i ′ ˙ u ij ′ ˙ u i ′ j ′ . Now summing over all indices j ′ yields: X i,j ′ ˙ u ij ′ = X i,j ′ ˙ u ij ′ + X i,i ′ : i = i ′ X j ′ ˙ u ij ′ ˙ u i ′ j ′ , where the last term on the right is equal to 0 thanks to (1.2). Thus X i,j ′ ˙ u ij ′ ( − ˙ u ij ′ ) ˙ u ij ′ = 0is a combination of positive operators, which can be equal to zero only if each of them is equal tozero, which amounts to: ˙ u ij ′ = ˙ u ij ′ for all i, j ′ , as desired. (cid:3) Remark . In the case d = 2, one can avoid the assumption of identical distribution of X i , byappropriately modifying the proof of Nica in the scalar case [Nic96, Theorem 5.1]. Indeed, therelation coming from κ O ( X ) n ( Y j , Y j , Y j ′ , . . . , Y j ′ ) = 0 can be written as κ n ( X ) u j u n − j ′ + κ n ( X ) u j u n − j ′ = 0 . Use (1.1) and multiply both sides of the above equation by u j ′ from the left to get: κ n ( X ) u j ′ u n − j ′ + κ n ( X ) u n − j ′ = κ n ( X ) u j ′ u n j ′ + κ n ( X ) u n +12 j ′ . Thanks to (1.2) this is equivalent to: κ n ( X ) u j ′ u n − j ′ = κ n ( X ) u n − j ′ ( u j ′ − ) − κ n ( X ) u j u j u n − j ′ . Now κ O ( X ) n ( Y j , Y j ′ , . . . , Y j ′ ) = 0 translates into κ n ( X ) u j u n − j ′ = − κ n ( X ) u j u n − j ′ , whose left-hand side appears as the last term in the above displayed equation. Substituting it yields κ n ( X ) u j ′ u n − j ′ = κ n ( X ) u n − j ′ ( u j ′ + u j − ) . And the term in paranthesis on the right-hand side vanish due to (1.1). Multiplying the equationby u j ′ from the left and using (1.1) to u j ′ on the left-hand side translates the relation to: κ n ( X ) u n − j ′ = κ n ( X ) u n j ′ where now only a cumulant of a single variable appears. Similarly one gets remaining relationswhich involve only κ n ( X ) or with u j ′ . All of them lead to the conclusion that either squares ofthe elements are all projections, or all cumulants κ n ( X i ) with n ≥ UANTUM BERNSTEIN’S THEOREM AND THE HYPEROCTAHEDRAL QUANTUM GROUP 9 Relevance of quantum Bernstein’s theorem
In classical Bernstein’s theorem and in its free version one cannot take a rotation which onlypermutes and changes signs of coordinates of the vector, as then independence or freeness ofcoordinates is trivially preserved. In other words the hyperoctahedral group always preserveindependence and freeness. In the quantum version the forbidden transformations come fromhyperoctahedral quantum group. We show that this assumption was indeed necessary: the hyper-octahedral quantum group always preserves freeness of coordinates. On the other hand our resultis closely related to Nica’s free Bernstein’s theorem, it is natural to ask whether it is a genuine ex-tension. We provide here examples of quantum subsets for which Nica’s theorem cannot be applied.3.1.
Preservation of freeness under transformations from the hyperoctahedral quan-tum group.
Let X = ( X , . . . , X d ) be a tuple consistsing of pairwise free entries belonging to asingle NCPS ( A , ϕ ). Our aim is to show that, whatever the marginal distributions of X are, thetuple Y = ( Y , . . . , Y d ) obtained by Y j = P di =1 X i ⊗ ˙ u ij ∈ A ⊗ O ( H + d ), is free with amalgamationover O ( H + d ) with respect to conditional expectation E = ϕ ⊗ id. To this end, we need to showthat all mixed O ( H + d )-valued free cumulants vanish. We calculate κ O ( H + d ) n ( Y j , . . . , Y j n ) = d X i ,...,i n =1 κ n ( X i , . . . , X i n ) ˙ u i j · · · ˙ u i n j n = d X i =1 κ n ( X i ) ˙ u ij · · · ˙ u ij n = 0where we used, respectively, (2.1), freeness of entries of X , and condition (3 ′ ) from the definitionof H + d together with the fact that the sequence j , . . . , j n is non-constant.3.2. Non-triviality of quantum Bernstein’s theorem.
To show that our quantum Bernstein’stheorem cannot be deduced from Nica’s free Bernstein’s theorem, we need to provide an exampleof a quantum subset X ⊂ O + d such that(1) X H + d (i.e. O ( O + d ) → O ( X ) cannot be factored through O ( H + d ));(2) X ∩ O + d ⊂ H d , i.e. the quotient of O ( X ) by the commutator ideal is a quotient of O ( H d ).In other words, the only points in X are those already in H d .The first item shows that such an X satisfies the assumption of our quantum Bernstein’s theorem,whereas (2) shows that there are too few points in X for Nica’s result to be applicable.Let G is a subgroup generated by X (in the sense of [BB10, SS16]). We expect that if X satisfies assumptions of our quantum Bernstein’s theorem (i.e. an identically distributed d -tupleof free random variables, after applying transformations from X , is free with amalgamation over O ( X )), then G also satisfies these assumptions. However, we were not able to find a proof of thisassertion without using the quantum Bernstein’s theorem itself: under these assumptions, thistuple is necessarily a semicircular system, and Curran showed in [Cur10, Proposition 3.5] that itsjoint distribution remains unchanged after applying transformations from O + d , so in particular itis preserved by transformations from G ⊂ O + d .With X being a quantum subset, the theorem is formally stronger than if we assumed that X is a subgroup. Note that the procedure of generation of quantum group from X can yield pointsthat are not obtained by generation of group from points of X , as discovered in [J´oz, Section 2.6].Because of that, and because of the statement from previous paragraph, it is desirable to providean example of X that is a quantum group from the very beginning.The quantum group which we need is O − ( d ). As noted in Remark 1.5, we have that H + d ⊂ O − ( d ), and by the results of [BBC07], we have that H + d = O − ( d ) for d ≥ O − ( d )is precisely H d : the conjunction of anticommutativity and commutativity in a single row/columnforces entries to satisfy the relation u ij u ij ′ = 0 = u ij u i ′ j , a defining relation for C ( H d ).Thus, to get X as described in first paragraph of the subsection, it is enough to take (for d ≥
3) any intermediate quantum space H + d ( X ⊆ O − ( d ), i.e. any intermediate quotient O ( O − ( d )) → O ( X ) → O ( H + d ), with the latter arrow being a proper surjection. In particular onecan take X = O − ( d ), which is itself a quantum group. Acknowledgement
The authors would like to thank Marek Bo˙zejko and Jacek Weso lowski for encouragement duringdevelopment of this project. We are also immensely grateful to Adam Skalski for his comments onan earlier version of the manuscript. PJ was partially supported by the NCN (National ScienceCenter) grant 2015/17/B/ST1/00085. KSz was partially supported by the NCN (National ScienceCenter) grant 2016/21/B/ST1/00005.
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Institute of Mathematics of the Polish Academy of Sciences, ul. ´Sniadeckich 8, 00–656 Warszawa,Poland and Institute of mathematics, University of Wroc law, pl. Grunwaldzki 2/4, 50–384 Wroc law,Poland
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