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Von Neumann-Jordan constant for Lebesgue-Bochner spaces. (English) Zbl 0912.46014
Summary: The von Neumann-Jordan (NJ-)constant for Lebesgue-Bochner spaces $$L_p(X)$$ is determined under some conditions on a Banach space $$X$$. In particular, the NJ-constant for $$L_r(c_p)$$ as well as $$c_p$$ (the space of $$p$$-Schatten class operators) is determined. For a general Banach space $$X$$, we estimate the NJ-constant of $$L_p(X)$$, which may be regarded as a sharpened result of a previous one concerning the uniform non-squareness for $$L_p(X)$$. Similar estimates are given for Banach sequence spaces $$\ell_p(X_i)$$ ($$\ell_p$$-sum of Banach spaces $$X_i$$), which gives a condition by NJ-constants of $$X_i$$’s under which $$\ell_p(X_i)$$ is uniformly non-square. A bi-product concerning ‘Clarkson’s inequality’ for $$L_p(X)$$ and $$\ell_p(X_i)$$ is also given.

MSC:
 46B20 Geometry and structure of normed linear spaces 46E40 Spaces of vector- and operator-valued functions 46B45 Banach sequence spaces
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