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On the von Neumann-Jordan constant
for Banach spaces

Authors: Mikio Kato and Yasuji Takahashi
Journal: Proc. Amer. Math. Soc. 125 (1997), 1055-1062
MSC (1991): Primary 46B20, 46B03, 46B42
MathSciNet review: 1371131
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Abstract: Let $C_{\mathrm {NJ}} (E)$ be the von Neumann-Jordan constant for a Banach space $E$. It is known that $1\le C_{\mathrm {NJ}}(E)\le 2$ for any Banach space $E$; and $E$ is a Hilbert space if and only if $C_{\mathrm {NJ}} (E)=1$. We show that: (i) If $E$ is uniformly convex, $C_{\mathrm {NJ}} (E)$ is less than two; and conversely the condition $C_{\mathrm {NJ}} (E)<2$ implies that $E$ admits an equivalent uniformly convex norm. Hence, denoting by $\widetilde C_{\mathrm {NJ}} (E)$ the infimum of all von Neumann-Jordan constants for equivalent norms of $E$, $E$ is super-reflexive if and only if $\widetilde C_{\mathrm {NJ}} (E)<2$. (ii) If $\widetilde C_{\mathrm {NJ}} (E)=2^{2/p-1}$, $1<p\le 2$ (the same value as that of $L_p$-space), $E$ is of Rademacher type $r$ and cotype $r'$ for any $r$ with $1\le r<p$, where $1/r+1/r'=1$; the converse holds if $E$ is a Banach lattice and $l_p$ is finitely representable in $E$ or $E'$.

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Additional Information

Mikio Kato
Affiliation: Department of Mathematics, Kyushu Institute of Technology, Tobata, Kitakyushu 804, Japan

Yasuji Takahashi
Affiliation: Department of System Engineering, Okayama Prefectural University, Soja 719-11, Japan

Keywords: von Neumann-Jordan constant, uniform convexity, super-reflexivity, type and cotype, finite representability, $p$-convexity and $p$-concavity for a Banach lattice
Received by editor(s): September 8, 1995
Additional Notes: The authors were supported in part by Grants-in-Aid for Scientific Research from the Ministry of Education, Science and Culture (07640225 (first author), 07640240 (second author))
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1997 American Mathematical Society