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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the von Neumann-Jordan constant for Banach spaces
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by Mikio Kato and Yasuji Takahashi
Proc. Amer. Math. Soc. 125 (1997), 1055-1062
DOI: https://doi.org/10.1090/S0002-9939-97-03740-4

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 $\tilde {C}_{\mathrm {NJ}}(E)$ the infimum of all von Neumann-Jordan constants for equivalent norms of $E$, $E$ is super-reflexive if and only if $\tilde {C}_{\mathrm {NJ}}(E)<2$. (ii) If $\tilde {C}_{\mathrm {NJ}}(E)=2^{2/p-1}$, $1<p\le 2$ (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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Bibliographic 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
  • 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
  • © Copyright 1997 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 125 (1997), 1055-1062
  • MSC (1991): Primary 46B20, 46B03, 46B42
  • DOI: https://doi.org/10.1090/S0002-9939-97-03740-4
  • MathSciNet review: 1371131