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## Studia Mathematica

2010 | 201 | 3 | 253-285
Tytuł artykułu

### Complex rotundities and midpoint local uniform rotundity in symmetric spaces of measurable operators

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EN
We investigate the relationships between strongly extreme, complex extreme, and complex locally uniformly rotund points of the unit ball of a symmetric function space or a symmetric sequence space E, and of the unit ball of the space E(ℳ,τ) of τ-measurable operators associated to a semifinite von Neumann algebra (ℳ,τ) or of the unit ball in the unitary matrix space $C_{E}$. We prove that strongly extreme, complex extreme, and complex locally uniformly rotund points x of the unit ball of the symmetric space E(ℳ,τ) inherit these properties from their singular value function μ(x) in the unit ball of E with additional necessary requirements on x in the case of complex extreme points. We also obtain the full converse statements for the von Neumann algebra ℳ with a faithful, normal, σ-finite trace τ as well as for the unitary matrix space $C_{E}$. Consequently, corresponding results on the global properties such as midpoint local uniform rotundity, complex rotundity and complex local uniform rotundity follow.
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Tom
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253-285
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wydano
2010
Twórcy
• Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, U.S.A.
autor
• Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152
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