As is well-known, on an Arens regular Banach algebra all continuous functionals are weakly almost periodic. In this paper, we show that ${\ell }^1$-bases which approximate upper and lower triangles of products of elements in the algebra produce large sets of functionals that are not weakly almost periodic. This leads to criteria for extreme non-Arens regularity of Banach algebras in the sense of Granirer. We find in particular that bounded approximate identities (bai's) and bounded nets converging to invariance (TI-nets) both fall into this approach, suggesting that this is indeed the main tool behind most known constructions of non-Arens regular algebras.

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关于巴拿赫代数的极端非阿伦斯正则性

众所周知，在阿伦斯正则巴拿赫代数上，所有连续泛函都是弱几乎周期性的。在本文中，我们表明${\ell }^1$- 近似代数中元素乘积的上下三角形的基产生大量几乎不是弱周期性的泛函。这导致了 Granirer 意义上的 Banach 代数的极端非阿伦斯正则性的标准。我们特别发现有界近似恒等式 (bai's) 和收敛到不变性的有界网 (TI-nets) 都属于这种方法，这表明这确实是大多数已知的非阿伦斯正则代数构造背后的主要工具。