Banach张量积中的von Neumann-Jordan常数

中文摘要 AI 辅助

我们研究了配备内射张量范数和射影张量范数的Banach张量积的von Neumann-Jordan常数。我们首先根据因子空间的几何性质建立下界估计，并证明当两个因子均为无限维时，两个典范张量积的von Neumann-Jordan常数均达到最大值2。对于有限维因子，我们获得了经典序列空间的显式结果，并通过算子空间和正交接触条件刻画了极值情形。我们进一步基于欧几里得截面导出了维数敏感的估计。最后，我们研究了等价重赋范下的常数，并确定了内射和射影张量积的精确重赋范包络。作为推论，我们刻画了这些张量积何时是超自反的以及何时是弱Hilbert空间。这些结果阐明了张量范数、维数和重赋范对Banach张量积的von Neumann-Jordan几何的影响。

英文摘要

We investigate the von Neumann--Jordan constant of Banach tensor products equipped with the injective and projective tensor norms. We first establish lower estimates in terms of the geometric properties of the factor spaces and show that, whenever both factors are infinite-dimensional, the von Neumann--Jordan constants of both canonical tensor products attain the maximal value \(2\). For finite-dimensional factors, we obtain explicit results for classical sequence spaces and characterize the extremal case through operator-space and orthogonal-contact conditions. We further derive dimension-sensitive estimates based on Euclidean sections. Finally, we study the constant under equivalent renormings and determine its exact renorming envelope for injective and projective tensor products. As consequences, we characterize when these tensor products are superreflexive and when they are weak Hilbert spaces. These results clarify the influence of tensor norms, dimension, and renorming on the von Neumann--Jordan geometry of Banach tensor products.

发表机构

• School of Mathematics and Statistics, Anqing Normal University（安庆师范大学数学与统计学院）
• Department of Mathematics, Sun Yat-sen University（中山大学数学系）

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