Conditional Expectations on Compact Quantum Groups and Quantum Double Cosets

Kalyuzhnyi, A. A.; Podkolzin, G. B.; Chapovskii, Yu. A.

doi:10.1007/s11253-005-0226-3

Cited by 3 publications

(6 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This brings a lot of trouble constructing non-trivial compact quantum hypergroups. Later on Kalyuzhnyi proposed [Kal01] a construction of compact quantum hypergroups using conditional expectations on compact quantum groups. The compact quantum hypergroups discussed in this paper mainly come from this construction.…”

Section: Consider An Elementmentioning

confidence: 99%

“…Theorem 1.3. [Kal01] Let G = (A, ∆, ǫ, S) be a compact quantum group. Let h be its Haar measure and P : A → B be an h-invariant conditional expectation that maps to a unital C * -subalgebra B of A.…”

Section: Consider An Elementmentioning

confidence: 99%

“…The original definition [CV99] is rather technical, which makes it difficult to construct examples. Thus relatively few concrete compact quantum hypergroups are known so far [CV99,Kal01]. This motivates us to present two approaches to constructions of compact quantum hypergroups in Section 2, to enlarge the class of relevant examples.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Infinitely divisible states on finite quantum groups

Zhang

2019

Math. Z.

View full text Add to dashboard Cite

In this paper we study the states of Poisson type and infinitely divisible states on compact quantum groups. Each state of Poisson type is infinitely divisible, i.e., it admits n-th root for all n ≥ 1. The main result is that on finite quantum groups infinitely divisible states must be of Poisson type. This generalizes Böge's theorem concerning infinitely divisible measures (commutative case) and Parthasarathy's result on infinitely divisible positive definite functions (cocommutative case). Two proofs are given.2010 Mathematics Subject Classification. Primary: 60B15, 60E07. Secondary: 16T20.

show abstract

Section: Consider An Elementmentioning

confidence: 99%

Section: Consider An Elementmentioning

confidence: 99%

See 1 more Smart Citation

Infinitely divisible states on finite quantum groups

Zhang

2019

Math. Z.

View full text Add to dashboard Cite

show abstract

“…The following construction, of new C * -hyperbialgebras from old, was described explicitly (in the context of compact quantum hypergroups) in the papers [10] and [11], but its origins go back much further (see [5] and references therein). All known examples of noncommutative C * -hyperbialgebras arise in this way from C * -bialgebras.…”

Section: * -Hyperbialgebras and The Conditional Expectation Construmentioning

confidence: 99%

“…The main results have been summarised in [15]. Recent years have also seen an increased interest in the noncommutative generalisation of classical hypergroups ( [3]), initiated by Chapovsky and Vainerman ( [5]) and continued, for example, in the papers [10] and [11]. Compact quantum hypergroups differ from compact quantum groups in that their coproduct need not be multiplicative.…”

mentioning

confidence: 99%

Completely positive quantum stochastic convolution cocycles and their dilations

Skalski¹

2007

Math. Proc. Camb. Phil. Soc.

View full text Add to dashboard Cite

Abstract. Stochastic generators of completely positive and contractive quantum stochastic convolution cocycles on a C * -hyperbialgebra are characterised. The characterisation is used to obtain dilations and stochastic forms of Stinespring decomposition for completely positive convolution cocycles on a C * -bialgebra.Stochastic (or Markovian) cocycles on operator algebras are basic objects of interest in quantum probability ([Acc]) and have been extensively investigated using quantum stochastic analysis (see [Lin]). There is also a well-developed theory of quantum Lévy processes, that is stationary, independent-increment, *-homomorphic processes on a *-bialgebra (see [Sch], [Fra] and references therein). Close examination of these two directions has naturally led to the notion of quantum stochastic convolution cocycle on a quantum group (or, more generally, on a coalgebra), as introduced and investigated in [ . Compact quantum hypergroups differ from compact quantum groups in that their coproduct need not be multiplicative. However, it remains completely positive, which makes compact quantum hypergroups, or more generally C * -hyperbialgebras, an appropriate category for the consideration of completely positive quantum stochastic convolution cocycles in a topological context (for the purely algebraic case see [FrS]). These cocycles may be viewed as natural counterparts of stationary, independent-increment processes on hypergroups. In [LS 4 ] it is shown that, under certain regularity conditions, they satisfy coalgebraic quantum stochastic differential equations.The aim of this paper is to prove dilation theorems for quantum stochastic convolution cocycles on a C * -bialgebra. To this end it is first necessary to establish the detailed structure of the stochastic generators of completely positive and contractive convolution cocycles. We give a direct derivation of this exploiting ideas used in the analysis of standard quantum stochastic cocycles with finite-dimensional noise space ([LiP]). Once the structure of generators is known, one may consider question of dilating completely positive convolution cocycles to * -homomorphic ones. In the context of standard quantum stochastic cocycles this problem was treated in [GLSW] and [GLW] (see also [Bel]). In the first of these papers it was shown

show abstract