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Freeness with amalgamation, limit theorems and S-transform in non-commutative probability spaces of type B

Mihai Popa (2010)

Colloquium Mathematicae

The paper addresses several problems left open by P. Biane, F. Goodman and A. Nica [Trans. Amer. Math. Soc. 355 (2003)]. The main result is that a type B non-commutative probability space can be studied in the framework of freeness with amalgamation. This view allows easy ways of constructing a version of the S-transform as well as proving analogues to the Central Limit Theorem and Poisson Limit Theorem.

From double Lie groups to quantum groups

Piotr Stachura (2005)

Fundamenta Mathematicae

It is shown, using geometric methods, that there is a C*-algebraic quantum group related to any double Lie group (also known as a matched pair of Lie groups or a bicrossproduct Lie group). An algebra underlying this quantum group is the algebra of a differential groupoid naturally associated with the double Lie group.

From Eckart and Young approximation to Moreau envelopes and vice versa

Jean-Baptiste Hiriart-Urruty, Hai Yen Le (2013)

RAIRO - Operations Research - Recherche Opérationnelle

In matricial analysis, the theorem of Eckart and Young provides a best approximation of an arbitrary matrix by a matrix of rank at most r. In variational analysis or optimization, the Moreau envelopes are appropriate ways of approximating or regularizing the rank function. We prove here that we can go forwards and backwards between the two procedures, thereby showing that they carry essentially the same information.

From isotonic Banach functionals to coherent risk measures

Zbigniew Dudek (2001)

Applicationes Mathematicae

Coherent risk measures [ADEH], introduced to study both market and nonmarket risks, have four characteristic properties that lead to the term “coherent” present in their name. Coherent risk measures regarded as functionals on the space L ∞ ( Ω , ℱ , ℙ ) have been extensively studied [De] with respect to these four properties. In this paper we introduce CRM functionals, defined as isotonic Banach functionals [Al], and use them to characterize coherent risk measures on the space L ∞ ( Ω , ℱ , ℙ ) as order opposites of CRM functionals....

From restricted type to strong type estimates on quasi-Banach rearrangement invariant spaces

María Carro, Leonardo Colzani, Gord Sinnamon (2007)

Studia Mathematica

Let X be a quasi-Banach rearrangement invariant space and let T be an (ε,δ)-atomic operator for which a restricted type estimate of the form ∥ T χ E ∥ X ≤ D ( | E | ) for some positive function D and every measurable set E is known. We show that this estimate can be extended to the set of all positive functions f ∈ L¹ such that | | f | | ∞ ≤ 1 , in the sense that ∥ T f ∥ X ≤ D ( | | f | | ₁ ) . This inequality allows us to obtain strong type estimates for T on several classes of spaces as soon as some information about the galb of the space X is known. In this paper...

From weak to strong types of L E 1 -convergence by the Bocce criterion

Erik Balder, Maria Girardi, Vincent Jalby (1994)

Studia Mathematica

Necessary and sufficient oscillation conditions are given for a weakly convergent sequence (resp. relatively weakly compact set) in the Bochner-Lebesgue space ℒ E 1 to be norm convergent (resp. relatively norm compact), thus extending the known results for ℒ ℝ 1 . Similarly, necessary and sufficient oscillation conditions are given to pass from weak to limited (and also to Pettis-norm) convergence in ℒ E 1 . It is shown that tightness is a necessary and sufficient condition to pass from limited to strong convergence....

Front d'onde analytique et décomposition microlocale des distributions

Pascal Laubin (1983)

Annales de l'institut Fourier

On étudie en détail une décomposition microlocale analytique de la distribution δ ( x - y ) suivant des distributions singulières en un seul point et dans une seule codirection. Cette décomposition est obtenue à partir d’opérateurs Fourier-Intégraux à phases complexes.On utilise ensuite cet outil pour démontrer le théorème de décomposition du front d’onde analytique des distributions. On établit également des théorèmes concernant la représentation globale des distributions comme sommes de valeurs au bord...

Fuglede-type decompositions of representations

Marek Kosiek (2002)

Studia Mathematica

It is shown that reducing bands of measures yield decompositions not only of an operator representation itself, but also of its commutant. This has many consequences for commuting Hilbert space representations and for commuting operators on Hilbert spaces. Among other things, it enables one to construct a Lebesgue-type decomposition of several commuting contractions without assuming any von Neumann-type inequality.

Full groups, flip conjugacy, and orbit equivalence of Cantor minimal systems

S. Bezuglyi, K. Medynets (2008)

Colloquium Mathematicae

We consider the full group [φ] and topological full group [[φ]] of a Cantor minimal system (X,φ). We prove that the commutator subgroups D([φ]) and D([[φ]]) are simple and show that the groups D([φ]) and D([[φ]]) completely determine the class of orbit equivalence and flip conjugacy of φ, respectively. These results improve the classification found in [GPS]. As a corollary of the technique used, we establish the fact that φ can be written as a product of three involutions from [φ].

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