  As is well-know, 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.
  These criteria can be applied to the main algebras in harmonic analysis such
as the group algebra, the measure algebra, the semigroup algebra (with certain
weights) and the Fourier algebra. In this paper, we apply our criteria to the
Lebesgue-Fourier algebra, the 1-Segal Fourier algebra and the Fig\`a-Talamanca
Herz algebra.
