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1

Geometric properties of noncommutative symmetric spaces of measurable operators and unitary matrix ideals

Czerwińska M., Kamińska A.

Commentationes Mathematicae

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2017

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Vol 57, No. 1

45--122

EN

This is a review article of geometric properties of noncommutative symmetric spaces of measurable operators E(M., t), where M is a semifinite von Neumann algebra with a faithful, normal, semifinite trace τ, and E is a symmetric function space. If E co is a symmetric sequence space then the analogous properties in the unitary matrix ideals CE are also presented. In the preliminaries we provide basic definitions and concepts illustrated by some examples and occasional proofs. In particular we list and discuss the properties of general singular value function, submajorization in the sense of Hardy, Littlewood and Polya, Kothe duality, the spaces Lp (M, τ), 1 ≤p < ∞, the identification of CE and G(B(H),tr) for some symmetric function space G, the commutative case when E is identified with E(N, t) for N isometric to L∞ with the standard integral trace, trace preserving *-isomorphisms between E and a *-subalgebra of E (M, τ), and a general method for removing the assumption of non-atomicity of . The main results on geometric properties are given in separate sections. We present the results on (complex) extreme points, (complex) strict convexity, strong extreme points and midpoint local uniform convexity, k-extreme points and k-convexity, (complex or local) uniform convexity, smoothness and strong smoothness, (strongly) exposed points, (uniform) Kadec-Klee properties, Banach-Saks properties, Radon-Nikodym property and stability in the sense of Krivine-Maurey. We also state some open problems.

2

On the strict convexity of Besicovitch-Musielak-Orlicz spaces of almost periodic functions equipped with the Orlicz norm

Daoui A., Morsli M., Smaali M.

Commentationes Mathematicae

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2013

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Vol. 53, No. 2

101--111

EN

We characterize the strict convexity of Besicovitch-Musielak-Orlicz spaces of almost periodic functions, when it is equipped with the Orlicz norm. It is shown that this property is equivalent to the strict convexity of the Musielak-Orlicz function generating the space.

3

Characterization of the uniform convexity of the Besicovitch-Musielak-Orlicz spaces of almost periodic functions

Morsli M., Smaali M.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2006

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Vol. 46, [Z] 2

215-231

EN

We introduce the new class of Besicovitch-Musielak-Orlicz spaces of almost periodic functions B'a.p..The uniform convexity of this space is characterized in terms of it’s generating functional [fi]

4

On Property β of Rolewicz in Köthe–Bochner Function Spaces

Kolwicz P.

Bulletin of the Polish Academy of Sciences. Mathematics

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2005

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Vol. 53, no 1

75-85

EN

It is proved that the Köthe–Bochner function space E(X) has property β if and only if X is uniformly convex and E has property β. In particular, property β does not lift from X to E(X) in contrast to the case of Köthe–Bochner sequence spaces.

5

Orthogonal uniform convexity in Köthe spaces and Orlicz spaces

Kolwicz P.

Bulletin of the Polish Academy of Sciences. Mathematics

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2002

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Vol. 50, no 4

395-411

EN

We study a geometric property in Köthe spaces which is called orthogonal uniform convexity (UC┴). It was introduced in [19]. We prove that the class of Köthe spaces with property (UC┴) is a proper subset of the class of uniformly monotone and P-convex Köthe spaces. Next we consider connections between (UC┴) and property (β) of Rolewicz. We shown that the implication (UC┴) → (β) is true in any Köthe sequence space. Moreover, we find criteria for Orlicz function (sequence) spaces to be orthogonally uniformly convex. As a corollary we get that there holds (UC) → (UC┴) → (β) in any Köthe sequence space and the converse of any of these implications is not true. Furthermore, the implications (UC) → (β) → (UC┴) hold in any Köthe function space and the second one cannot be reversed.

6

On uniform convexity and modular approximation in the Weyl-Orlicz spaces

Bedouhene F., Morsli M.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2001

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[Z] 41

25-39

EN

Uniform convexity is well characterized in Orlicz spaces for both the Luxemburg and Orlicz norms [2], [3], [4] and in Besicovitch-Orlicz spaces of almost periodic functions for the Luxemburg norm [7]. The existence problem of the projection operator on closed convex subsets was considered in [5] in the case of Orlicz spaces, and in [8] in the case of Besicovitch-Orlicz space of almost periodic functions. In this paper we characterize the uniform convexity property in the Weyl-Orlicz spaces of almost periodic functions (W^a.p.) under the Luxemburg norm. Next this result is used to state the existence of the projection operator on complete convex subsets of W^a.p.

7

Some sequence spaces defined by [N,Pn] summability

Bhardwaj V., Singh N.

Demonstratio Mathematica

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1999

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Vol. 32, nr 3

539-546

EN

The object of this paper is to introduce some new sequence spaces which arise from the notion of [N, pn] summability. Some topological results, certain inclusion relations and a result on matrix transformations have been discussed.

8

On uniform convexity of some spaces of multifunctions

Kasperski A.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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1999

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[Z] 39

87-95

EN

We introduce the appropriate space of multifunctions and the notions of uniform convexity of some subsets of this space. We get some remarks and theorems on uniform convexity of some subsets of this space. We prove that Musielak-Orlicz space of multifunctions is uniformly convex if the as-sumptions of Theorem 11.6 from [6] hold.