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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.

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