Positive Definite Functions on Coxeter Groups with Applications to Operator Spaces and Noncommutative Probability

Bożejko, Marek; Gal, Światosław R.; Młotkowski, Wojciech

doi:10.1007/s00220-018-3160-6

Cited by 10 publications

(6 citation statements)

References 29 publications

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“…Apparently no version of Drury's theorem is known for these notions in non-abelian groups. We refer to Bożejko and Speicher's [6] and also the recent work [5] for some results on completely positive functions on Coxeter groups that may be related to our own. See also [4].…”

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confidence: 87%

Completely Sidon sets in $C^*$-algebras (New title)

Pisier¹

2017

Preprint

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A sequence in a C * -algebra A is called completely Sidon if its span in A is completely isomorphic to the operator space version of the space ℓ 1 (i.e. ℓ 1 equipped with its maximal operator space structure). The latter can also be described as the span of the free unitary generators in the (full) C * -algebra of the free group F ∞ with countably infinitely many generators. Our main result is a generalization to this context of Drury's classical theorem stating that Sidon sets are stable under finite unions. In the particular case when A = C * (G) the (maximal) C * -algebra of a discrete group G, we recover the non-commutative (operator space) version of Drury's theorem that we recently proved. We also give several non-commutative generalizations of our recent work on uniformly bounded orthonormal systems to the case of von Neumann algebras equipped with normal faithful tracial states. MSC Classif. 43A46, 46L06Recently, following the impulse of Bourgain and Lewko [1], we studied in [17] the uniformly bounded orthonormal systems that span in L ∞ a subspace isomorphic to ℓ 1 by the basis to basis equivalence, and we called them Sidon sequences in analogy with the case of characters on compact abelian groups. One of the main results in [17] says that if a uniformly bounded orthonormal system {ψ n } in L 2 of a probability space (Ω, P) is the union {ψ 1 n } ∪ {ψ 2 n } of two Sidon sequences, then the sequence {ψ n ⊗ ψ n ⊗ ψ n ⊗ ψ n } or simply {ψ ⊗ 4 n } is Sidon in L ∞ (Ω 4 , P 4 ). Our goal in this paper is to generalize this result to sequences in a non-commutative C * -algebra. The central ingredient of our method in [17] is the spectral decomposition of the Ornstein-Uhlenbeck semigroup for a Gaussian measure on R n . Since this has all sorts of non-commutative analogues, it is natural to try to extend the results of [17] to non-commutative von Neumann or C * -algebras in place of L ∞ . In [17] "subgaussian" and "randomly Sidon" sequences play an important role. Although the non-commutative analogue of a subgaussian system is not clear (see however Remark 4.10), and that of "randomly Sidon set" eludes us for the moment, we are able in the present paper to extend several of the main results of [17], in particular we recover an analogue of Drury's famous union theorem for Sidon sets in groups. In the commutative case the fundamental example of Sidon set is the set formed of the canonical generators in the group Z ∞ formed of all the finitely supported functions f : N → Z. This is sometimes referred to as the free Abelian group with countably infinitely many generators. The dual of the discrete group Z ∞ is the compact group T N , and the von Neumann algebra of Z ∞ can be identified with L ∞ (T N ). The analogue of this for our work is the free group F ∞ with countably infinitely many generators, and its von Neumann algebra M F∞ . In the commutative case the generators of Z ∞ correspond in L ∞ (T N ) to independent random variables uniformly distributed over T. In classical Sidon set theory, the associated Riesz pr...

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confidence: 87%

Completely Sidon sets in $C^*$-algebras (New title)

Pisier¹

2017

Preprint

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“…In [11] it is proved that the generators in any Coxeter group satisfy the weak Sidon property and the preceding remark is explicitly applied to that case.…”

Section: λ(P)-setsmentioning

confidence: 99%

Completely Sidon sets in discrete groups

Pisier¹

2020

Annali Scuola Normale Superiore - Classe Di Scienze

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A subset of a discrete group G is called completely Sidon if its span in C * (G) is completely isomorphic to the operator space version of the space ℓ 1 (i.e. ℓ 1 equipped with its maximal operator space structure). We recently proved a generalization to this context of Drury's classical union theorem for Sidon sets: completely Sidon sets are stable under finite unions. We give a different presentation of the proof emphasizing the "interpolation property" analogous to the one Drury discovered. In addition we prove the analogue of the Fatou-Zygmund property: any bounded Hermitian function on a symmetric completely Sidon set Λ ⊂ G \ {1} extends to a positive definite function on G. In the final section, we give a completely isomorphic characterization of the closed span C Λ of a completely Sidon set in C * (G): the dual (in the operator space sense) of C Λ is exact iff Λ is completely Sidon. In particular, Λ is completely Sidon as soon as C Λ is completely isomorphic (by an arbitrary isomorphism) to ℓ 1 (Λ) equipped with its maximal operator space structure.

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“…They showed that it implies that infinite Coxeter groups do not have Kazhdan's property (T ). This result was further generalized to multi-parameters [BS03] and also other variants of the Coxeter function (colour-length) were studied [BS96,BGM18]. All the considered functions share two distinctive features: (I) they are positive definite on the continuos set −1 ≤ q ≤ 1, (II) they are not (generically) invariant by conjugation.…”

Section: Introductionmentioning

confidence: 96%

“…Note that the set of parameters for which the signed reflection function is positive definite is a discrete set (except the degenerate case q + = 0), which confirms that the behaviour of the reflection function is very different then its Coxeter counterpart and that the properties (I) and (II) are correlated. Another difference is visible in the strong correlation between parameters q + and q − , while the parameters in the multivariate versions of the Coxeter function studied in [BS03,BGM18] can take any value in the interval [−1, 1] independently. The methods used in the previously mentioned works on the Coxeter functions are not applicable here and our approach is based on the connection between the representation theory of the hyperoctahedral group and symmetric functions.…”

Section: Introductionmentioning

confidence: 99%

Reflection length with two parameters in the asymptotic representation theory of type B/C and applications

Bożejko¹,

Dołęga²,

Ejsmont³

et al. 2021

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We introduce a two-parameter function φ q+,q− on the infinite hyperoctahedral group, which is a bivariate refinement of the reflection length keeping track of the long and the short reflections separately. We provide a complete characterization of the parameters q + , q − when the signed reflection function φ q+,q− is positive definite and we prove that this condition holds if and only if φ q+,q− is an extreme character of the infinite hyperoctahedral group. We construct the corresponding representations as a natural action of the hyperoctahedral group B(n) on the tensor product of n copies of a vector space, which gives a two-parameter analog of the classical construction of Schur-Weyl.We apply our characterization to construct a cyclic Fock space of type B which generalizes the one-parameter construction in type A found previously by Bożejko and Guta. We also construct a new cyclic Gaussian operator of type B and we relate its moments with the Askey-Wilson-Kerov distribution by using the notion of cycles on pair-partitions, which we introduce here.

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Positive Definite Functions on Coxeter Groups with Applications to Operator Spaces and Noncommutative Probability

Cited by 10 publications

References 29 publications

Completely Sidon sets in $C^*$-algebras (New title)

Completely Sidon sets in $C^*$-algebras (New title)

Completely Sidon sets in discrete groups

Reflection length with two parameters in the asymptotic representation theory of type B/C and applications

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