Smooth and strongly smooth points in symmetric spaces of measurable operators

Smooth and strongly smooth points in symmetric spaces of measurable operators We investigate the relationships between smooth and strongly smooth points of the unit ball of an order continuous symmetric function space E, and of the unit ball of the space of τ-measurable operators $${E(\mathcal{M},\tau)}$$ associated to a semifinite von Neumann algebra $${(\mathcal{M}, \tau)}$$ . We prove that x is a smooth point of the unit ball in $${E(\mathcal{M}, \tau)}$$ if and only if the decreasing rearrangement μ(x) of the operator x is a smooth point of the unit ball in E, and either μ(∞; f) = 0, for the function $${f\in S_{E^{\times}}}$$ supporting μ(x), or s(x *) = 1. Under the assumption that the trace τ on $${\mathcal{M}}$$ is σ-finite, we show that x is strongly smooth point of the unit ball in $${E(\mathcal{M}, \tau)}$$ if and only if its decreasing rearrangement μ(x) is a strongly smooth point of the unit ball in E. Consequently, for a symmetric function space E, we obtain corresponding relations between smoothness or strong smoothness of the function f and its decreasing rearrangement μ(f). Finally, under suitable assumptions, we state results relating the global properties such as smoothness and Fréchet smoothness of the spaces E and $${E(\mathcal{M},\tau)}$$ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Smooth and strongly smooth points in symmetric spaces of measurable operators

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Publisher
SP Birkhäuser Verlag Basel
Copyright
Copyright © 2011 by Springer Basel AG
Subject
Mathematics; Potential Theory; Operator Theory; Fourier Analysis; Econometrics; Calculus of Variations and Optimal Control; Optimization
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-010-0108-2
Publisher site
See Article on Publisher Site

Abstract

We investigate the relationships between smooth and strongly smooth points of the unit ball of an order continuous symmetric function space E, and of the unit ball of the space of τ-measurable operators $${E(\mathcal{M},\tau)}$$ associated to a semifinite von Neumann algebra $${(\mathcal{M}, \tau)}$$ . We prove that x is a smooth point of the unit ball in $${E(\mathcal{M}, \tau)}$$ if and only if the decreasing rearrangement μ(x) of the operator x is a smooth point of the unit ball in E, and either μ(∞; f) = 0, for the function $${f\in S_{E^{\times}}}$$ supporting μ(x), or s(x *) = 1. Under the assumption that the trace τ on $${\mathcal{M}}$$ is σ-finite, we show that x is strongly smooth point of the unit ball in $${E(\mathcal{M}, \tau)}$$ if and only if its decreasing rearrangement μ(x) is a strongly smooth point of the unit ball in E. Consequently, for a symmetric function space E, we obtain corresponding relations between smoothness or strong smoothness of the function f and its decreasing rearrangement μ(f). Finally, under suitable assumptions, we state results relating the global properties such as smoothness and Fréchet smoothness of the spaces E and $${E(\mathcal{M},\tau)}$$ .

Journal

PositivitySpringer Journals

Published: Feb 5, 2011

References

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