Bulletin of the
Korean Mathematical Society BKMS

Let ${\mathcal A}$ be a Banach algebra, we say that ${\mathcal A}$ has the strongly double limit property (SDLP) if for each bounded net $(a_{\alpha})$ in ${\mathcal A}$ and each bounded net $(a^*_{\beta})$ in ${\mathcal A}^*$, $\lim_{\alpha}\lim_{\beta} \left\langle a_{\alpha} , a^*_{\beta}\right\rangle = \lim_{\beta} \lim_{\alpha} \left\langle a_{\alpha} , a^*_{\beta} \right\rangle $ whenever both iterated limits exist. In this paper among other results we show that if ${\mathcal A}$ has the SDLP and ${\mathcal A}^{**}$ is $(n-2)$-weakly amenable, then ${\mathcal A}$ is $n$-weakly amenable. In particular, it is shown that if ${\mathcal A}^{**}$ is weakly amenable and ${\mathcal A}$ has the SDLP, then ${\mathcal A}$ is weakly amenable.

Keywords: Banach algebra, weak amenability, Arens regular, $n$-weak amenability

MSC numbers: 46H20, 46H40