Schoenberg correspondence for multifaced independence

Gerhold, Malte

doi:10.48550/arxiv.2104.02985

Cited by 5 publications

(9 citation statements)

References 17 publications

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“…The axiomatization of the concept of independence in noncommutative probability based on universal products (suitable replacements for the product measure in classical probability) has been initiated by Ben Ghorbal and Schürmann in [BGS02]. The different independences focused on in the article at hand -bifree, bimonotone, and biBoolean independence -are covered by a multivariate extension of the theory of universal products which was spelled out explicitly by Manzel and Schürmann in [MS17], see also [Ger21,Section 3]. Those three independences can be derived from Gu and Skoufranis' c-bifree independence [GS17] very much like free, monotone and boolean independence are derived from Speicher and Bożejko's c-free independence [BS91].…”

Section: Combinatorics Of Two-faced Independencesmentioning

confidence: 99%

“…A unified framework for cumulants with respect to universal product independences is given by Manzel and Schürmann [MS17]. In this case, the moment-cumulant relations can always be expressed via a Hopf-algebraic exponential and logarithm, see also [Ger21]. Formulas relating different sets of noncommutative cumulants of a random variable were proved in [AHLV15] and are still subject of great attention.…”

Section: Notationmentioning

confidence: 99%

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Shuffle Algebras and Non-Commutative Probability for Pairs of Faces

Diehl¹,

Gerhold²,

Gilliers³

2022

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One can build an operatorial model for freeness by considering either the righthanded either the left-handed representation of algebras of operators acting on the free product of the underlying pointed Hilbert spaces. Considering both at the same time, that is computing distributions of operators in the algebra generated by the left-and right-handed representations, led Voiculescu in 2013 to define and study bifreeness and, in the sequel, triggered the development of an extension of noncommutative probability now frequently referred to as multi-faced (two-faced in the example given above). Many examples of two-faced independences emerged these past years. Of great interest to us are biBoolean, bifree and Type I bimonotone independences. In this paper we extend the preLie calculus prevailing to free, Boolean and monotone moment-cumulant relations initiated by K. Ebrahimi-Fard and F. Patras to their above mentioned two-faced equivalents. M.G. is supported by the German Research Foundation (DFG) grant no. 397960675. J.D. and N.G. are supported by the trilateral (DFG/ANR/JST) grant "EnhanceD Data stream Analysis with the Signature Method" . N.G is supported by ANR "STARS".

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Section: Combinatorics Of Two-faced Independencesmentioning

confidence: 99%

Section: Notationmentioning

confidence: 99%

Shuffle Algebras and Non-Commutative Probability for Pairs of Faces

Diehl¹,

Gerhold²,

Gilliers³

2022

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“…To begin, as * -algebras over C are not assumed to be unital, one needs to be careful about positivity. The following definition is based on [Ger21,Lac15]. Note that the term "state" is called "strongly positive linear functional" in [Lac15] and "restricted state" in [Ger21], but we simply call it "state" in this paper.…”

Section: Setup Notation and Remarksmentioning

confidence: 99%

“…This means that if we drop the positivity from the definition of independence, then we get various more notions of multi-faced independence. An axiomatic (or categorical) formulation of multi-faced independence is discussed in [MS17] (see also [GLS16], [Ger21,Definition 3.3] or Definitions 4.1 and 4.3 below), which includes all the above mentioned examples. 1.2.…”

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confidence: 99%

Towards a Classification of Multi-Faced Independence: A Representation-Theoretic Approach

Gerhold¹,

Hasebe²,

Ulrich³

2021

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We attack the classification problem of multi-faced independences, the first non-trivial example being Voiculescu's bi-freeness. While the present paper does not achieve a complete classification, it formalizes the idea of lifting an operator on a pre-Hilbert space in a "universal" way to a larger product space, which is key for the construction of (old and new) examples. It will be shown how universal lifts can be used to construct very well-behaved (multi-faced) independences in general. Furthermore, we entirely classify universal lifts to the tensor product and to the free product of pre-Hilbert spaces. Our work brings to light surprising new examples of 2-faced independences. Most noteworthy, for many known 2-faced independences, we find that they admit continuous deformations within the class of 2-faced independences, showing in particular that, in contrast with the single faced case, this class is infinite (and even uncountable).

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“…[6,7]). Also, following Voiculescu's invention of bifreeness [44], many examples of multivariate independences have been exhibited [13,16,21,22,23,29,30] and some general theory has been developed [14,32], and all those independences have associated classes of Lévy processes that are very interesting to study. The main aim of these notes is to give a unified approach to these different situations.…”

Section: Introductionmentioning

confidence: 99%

Categorial Independence and Lévy Processes

Gerhold¹,

Lachs²,

Schürmann³

2022

SIGMA

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We generalize Franz' independence in tensor categories with inclusions from two morphisms (which represent generalized random variables) to arbitrary ordered families of morphisms. We will see that this only works consistently if the unit object is an initial object, in which case the inclusions can be defined starting from the tensor category alone. The obtained independence for morphisms is called categorial independence. We define categorial Lévy processes on every tensor category with initial unit object and present a construction generalizing the reconstruction of a Lévy process from its convolution semigroup via the Daniell-Kolmogorov theorem. Finally, we discuss examples showing that many known independences from algebra as well as from (noncommutative) probability are special cases of categorial independence.

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Schoenberg correspondence for multifaced independence

Cited by 5 publications

References 17 publications

Shuffle Algebras and Non-Commutative Probability for Pairs of Faces

Shuffle Algebras and Non-Commutative Probability for Pairs of Faces

Towards a Classification of Multi-Faced Independence: A Representation-Theoretic Approach

Categorial Independence and Lévy Processes

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