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Metric Entropy of Homogeneous Spaces

Stanisław Szarek (1998)

Banach Center Publications

For a precompact subset K of a metric space and ε > 0, the covering number N(K,ε) is defined as the smallest number of balls of radius ε whose union covers K. Knowledge of the metric entropy, i.e., the asymptotic behaviour of covering numbers for (families of) metric spaces is important in many areas of mathematics (geometry, functional analysis, probability, coding theory, to name a few). In this paper we give asymptotically correct estimates for covering numbers for a large class of homogeneous...

Monotone convolution semigroups

Takahiro Hasebe (2010)

Studia Mathematica

We study how a property of a monotone convolution semigroup changes with respect to the time parameter. Especially we focus on "time-independent properties": in the classical case, there are many properties of convolution semigroups (or Lévy processes) which are determined at an instant, and moreover, such properties are often characterized by the drift term and Lévy measure. In this paper we exhibit such properties of monotone convolution semigroups; an example is the concentration of the support...

Multiplicative monotone convolutions

Uwe Franz (2006)

Banach Center Publications

Recently, Bercovici has introduced multiplicative convolutions based on Muraki's monotone independence and shown that these convolution of probability measures correspond to the composition of some function of their Cauchy transforms. We provide a new proof of this fact based on the combinatorics of moments. We also give a new characterisation of the probability measures that can be embedded into continuous monotone convolution semigroups of probability measures on the unit circle and briefly discuss...

New Examples of Convolutions and Non-Commutative Central Limit Theorems

Marek Bożejko, Janusz Wysoczański (1998)

Banach Center Publications

A family of transformations on the set of all probability measures on the real line is introduced, which makes it possible to define new examples of convolutions. The associated central limit theorems are studied, and examples of the limit measures, related to the classical, free and boolean convolutions, are shown.

Non-commutative entropy computations for continuous fields and cross-products

Emmanuel Germain (2007)

Annales mathématiques Blaise Pascal

We present here two non-commutative situations where dynamical entropy estimates are possible. The first result is concerned with automorphisms of cross-products by an exact group that commute with the group action and generalizes the result known for amenable groups. The second is about continuous fields of C * -algebras and C ( X ) -automorphisms. Each result relies on explicit factorization via matrices.

Noncommutative extensions of the Fourier transform and its logarithm

Romuald Lenczewski (2002)

Studia Mathematica

We introduce noncommutative extensions of the Fourier transform of probability measures and its logarithm to the algebra (S) of complex-valued functions on the free semigroup S = FS(z,w) on two generators. First, to given probability measures μ, ν with all moments finite, we associate states μ̂, ν̂ on the unital free *-bialgebra (ℬ,ε,Δ) on two self-adjoint generators X,X’ and a projection P. Then we introduce and study cumulants which are additive under the convolution μ̂* ν̂ = μ̂ ⊗ ν̂ ∘ Δ when...

Noncommutative fractional integrals

Narcisse Randrianantoanina, Lian Wu (2015)

Studia Mathematica

Let ℳ be a hyperfinite finite von Nemann algebra and ( ℳ k ) k ≥ 1 be an increasing filtration of finite-dimensional von Neumann subalgebras of ℳ. We investigate abstract fractional integrals associated to the filtration ( ℳ k ) k ≥ 1 . For a finite noncommutative martingale x = ( x k ) 1 ≤ k ≤ n ⊆ L ₁ ( ℳ ) adapted to ( ℳ k ) k ≥ 1 and 0 < α < 1, the fractional integral of x of order α is defined by setting I α x = ∑ k = 1 n ζ k α d x k for an appropriate sequence ( ζ k ) k ≥ 1 of scalars. For the case of a noncommutative dyadic martingale in L₁() where is the type II₁ hyperfinite factor equipped...

Noncommutative independence in the infinite braid and symmetric group

Rolf Gohm, Claus Köstler (2011)

Banach Center Publications

This is an introductory paper about our recent merge of a noncommutative de Finetti type result with representations of the infinite braid and symmetric group which allows us to derive factorization properties from symmetries. We explain some of the main ideas of this approach and work out a constructive procedure to use in applications. Finally we illustrate the method by applying it to the theory of group characters.

Noncommutative weak Orlicz spaces and martingale inequalities

Turdebek N. Bekjan, Zeqian Chen, Peide Liu, Yong Jiao (2011)

Studia Mathematica

This paper is devoted to the study of noncommutative weak Orlicz spaces and martingale inequalities. The Marcinkiewicz interpolation theorem is extended to include noncommutative weak Orlicz spaces as interpolation classes. As an application, we prove the weak type Φ-moment Burkholder-Gundy inequality for noncommutative martingales through establishing a weak type Φ-moment noncommutative Khinchin inequality for Rademacher random variables.

On 0 - 1 measure for projectors

Václav Alda (1980)

Aplikace matematiky

An example of a finite set of projectors in E 3 is exhibited for which no 0-1 measure exists.

Currently displaying 81 – 100 of 184