Wyniki wyszukiwania

1

Domination of operators in the non-commutative setting

100%

Oikhberg T., Spinu E.

Studia Mathematica

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2013

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tom 219

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nr 1

35-67

EN

We consider majorization problems in the non-commutative setting. More specifically, suppose E and F are ordered normed spaces (not necessarily lattices), and 0 ≤ T ≤ S in B(E,F). If S belongs to a certain ideal (for instance, the ideal of compact or Dunford-Pettis operators), does it follow that T belongs to that ideal as well? We concentrate on the case when E and F are C*-algebras, preduals of von Neumann algebras, or non-commutative function spaces. In particular, we show that, for C*-algebras 𝓐 and ℬ, the following are equivalent: (1) at least one of the two conditions holds: (i) 𝓐 is scattered, (ii) ℬ is compact; (2) if 0 ≤ T ≤ S : 𝓐 → ℬ, and S is compact, then T is compact.

2

Some results on metric trees

100%

Aksoy A., Oikhberg T.

Banach Center Publications

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2010

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tom 91

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nr 1

9-34

EN

Using isometric embedding of metric trees into Banach spaces, this paper will investigate barycenters, type and cotype, and various measures of compactness of metric trees. A metric tree (T, d) is a metric space such that between any two of its points there is a unique arc that is isometric to an interval in ℝ. We begin our investigation by examining isometric embeddings of metric trees into Banach spaces. We then investigate the possible images x₀ = π((x₁ + ... + xₙ)/n), where π is a contractive retraction from the ambient Banach space X onto T (such a π always exists) in order to understand the "metric" barycenter of a family of points x₁,...,xₙ in a tree T. Further, we consider the metric properties of trees such as their type and cotype. We identify various measures of compactness of metric trees (their covering numbers, ϵ-entropy and Kolmogorov widths) and the connections between them. Additionally, we prove that the limit of the sequence of Kolmogorov widths of a metric tree is equal to its ball measure of non-compactness.

3

Embeddings of finite-dimensional operator spaces into the second dual

100%

Arias A., Oikhberg T.

Studia Mathematica

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2007

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tom 181

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nr 2

181-198

EN

We show that, if a a finite-dimensional operator space E is such that X contains E C-completely isomorphically whenever X** contains E completely isometrically, then E is $2^{15} C^{11}$-completely isomorphic to Rₘ ⊕ Cₙ for some n, m ∈ ℕ ∪ {0}. The converse is also true: if X** contains Rₘ ⊕ Cₙ λ-completely isomorphically, then X contains Rₘ ⊕ Cₙ (2λ + ε)-completely isomorphically for any ε > 0.

Ograniczanie wyników

2 Studia Mathematica

1 Banach Center Publications

3 Oikhberg T.

1 Aksoy A.

1 Arias A.

1 Spinu E.

1 2013

1 2010

1 2007

Domination of operators in the non-commutative setting

Some results on metric trees

Embeddings of finite-dimensional operator spaces into the second dual