Complex locally uniform rotundity of Musielak-Orlicz spaces

Wang, Tingfu; Teng, Yanmei

doi:10.1007/bf02876036

Cited by 8 publications

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“…In 2000, T. Wang and Y. Teng [109] defined C − LU R points and C − LU R spaces and obtained criteria for this property in the class of Musielak-Orlicz spaces of vector-valued functions. A point x ∈ S X , where (X, · ) is a complex normed space, is a point of complex local uniform convexity (C − LU R point) [109] if for every ǫ > 0 there exists δ(x, ǫ) > 0 such that sup λ=±1,±i…”

Section: Complex Local Uniform Convexitymentioning

confidence: 99%

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

Czerwińska

Kamińska

2017

commath

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Abstract. This is a survey article of geometric properties of noncommutative symmetric spaces of measurable operators E(M, τ ), where M is a semifinite von Neumann algebra with a faithful, normal, semifinite trace τ , and E is a symmetric function space. If E ⊂ c0 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 Pólya, Köthe duality, the spaces Lp (M, τ ), 1 ≤ p < ∞, the identification between CE and G(B(H), tr) for some symmetric function space G, the commutative case when E is identified with E(N , τ ) 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 of removing the assumption of non-atomicity of M. 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-Nikodým property and stability in the sense of Krivine-Maurey. We also state some open problems.In 1937, John von Neumann [83] pp. 205-218, observed that for a symmetric norm · in R n , it is possible to define a norm on the space of n × n matrices x by setting x = {s i (x)} n i=1 , where s i (x), i = 1, 2 . . . , n, are eigenvalues of the matrix |x| = (x * x) 1/2 ordered in a decreasing manner. Later on in the forties and fifties, J. von Neumann and R. Schatten developed analogous theory for infinite dimensional compact operators. They defined and studied unitary matrix ideals C E corresponding to a symmetric sequence Banach space (E, · E ). The space consists of all compact operators x on a Hilbert space such that {s n (x)} ⊂ E with the norm x = {s n (x)} E , where s n (x), n ∈ N, are singular numbers of x, that is eigenvalues of |x|. For E = ℓ 1 , the space C E is called the trace class of operators or the space of nuclear operators, while if E = ℓ 2 then it is called the class of Hilbert-Schmidt operators. The first monograph of these spaces was written by R. Schatten in 1960 [93], and later on in 1969 by I. C. Gohberg and M. G. Krein [49]. In 1967, C. McCarthy wrote an article on the now called the Schatten classes C p , 0 < p ≤ ∞, that is the spaces C E when E = ℓ p , and showed among others that this space is uniformly convex for 1 < p < ∞ [78]. The beginning of the theory of symmetric spaces of measurable operators can be traced back to the early fifties. It was then when I. Segal and J. Dixmier [94,29] laid out the foundation for noncommutative L p (M, τ ) spaces, 0 < p < ∞, by introducing t...

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Section: Complex Local Uniform Convexitymentioning

confidence: 99%

“…A point x ∈ S X , where (X, · ) is a complex normed space, is a point of complex local uniform convexity (C − LU R point) [109] if for every ǫ > 0 there exists δ(x, ǫ) > 0 such that sup λ=±1,±i…”

Section: Complex Local Uniform Convexitymentioning

confidence: 99%

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

Czerwińska

Kamińska

2017

commath

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“…Globevnik [13] introduced the notion of uniform C-rotundity and showed that the complex space L 1 ( , µ) has this property. Next Wang and Teng [27] introduced the notion of locally uniform C-rotundity, obtaining criteria for this property in the class of Musielak-Orlicz spaces.…”

Section: The Space (E C ) Is Called In This Paper a Complex Köthe Smentioning

confidence: 99%

Relationships between monotonicity and complex rotundity properties with some consequences

Hudzik¹,

Narloch²

2005

MATH. SCAND.

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It is proved that a point f of the complexification E C of a real Köthe space E is a complex extreme point if and only if |f | is a point of upper monotonicity in E. As a corollary it follows that E is strictly monotone if and only if E C is complex rotund. It is also shown that E is uniformly monotone if and only if E C is uniformly complex rotund. Next, the fact that |x| ∈ S(E + ) is a ULUM-point of E whenever x is a C-LUR-point of S(E C ) is proved, whence the relation that E is a ULUM-space whenever E C is C-LUR is concluded. In the second part of this paper these general results are applied to characterize complex rotundity of properties Calderón-Lozanovskiȋ spaces, generalized Calderón-Lozanovskiȋ spaces and Orlicz-Lorentz spaces. IntroductionFirst we introduce the notations and define the notions used in this paper. Let (T , , µ) be a complete and σ -finite measure space and L 0 = L 0 (T , , µ) be the space of all (equivalence classes of) -measurable real functions defined on T . A Banach space (E, E ) is said to be a real Köthe space if E ⊂ L 0 and: (i) for every x ∈ L 0 and y ∈ E with |x(t)| |y(t)| µ-a.e. in T , we havex ∈ E and x E y E ,(ii) there is a function x ∈ E such that x(t) > 0 for any t ∈ T .In the whole paper, if a real Köthe space E is fixed, S(E) and B(E) denote the unit sphere and the unit ball of E, respectively. The same notations are used for complex Köthe spaces defined bellow. In this paper global and local monotonicity properties are considered for real Köthe spaces only although these properties can be also defined for abstract Banach lattices (see [2]). We say that E is strictly monotone (E ∈ (SM) for short) if for every x, y in the positive cone E + = {x ∈ E : x 0} we have x E < y E whenever x y and x = y. E is said to be uniformly monotone (E ∈ (UM))

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“…Further details about the geometry of complex Banach spaces can be found in [10,[16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

Complex rotundity of Orlicz sequence spaces equipped with the p-Amemiya norm

Chen

Cui

2011

Journal of Mathematical Analysis and Applications

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In this paper, two equivalent definitions of complex strongly extreme points in general complex Banach spaces are shown. It is proved that for any Orlicz sequence space equipped with the p-Amemiya norm (1 p < ∞, p is odd), complex strongly extreme points of the unit ball coincide with complex extreme points of the unit ball. Moreover, criteria for them in Orlicz sequence spaces equipped with the p-Amemiya norm are given. Criteria for complex mid-point locally uniform rotundity and complex rotundity of Orlicz sequence spaces equipped with the p-Amemiya norm are also deduced.

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Complex locally uniform rotundity of Musielak-Orlicz spaces

Cited by 8 publications

References 1 publication

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

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

Relationships between monotonicity and complex rotundity properties with some consequences

Complex rotundity of Orlicz sequence spaces equipped with the p-Amemiya norm

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