$n$-weak amenability and strong double limit property
Bull. Korean Math. Soc. 2005 Vol. 42, No. 2, 359-367
Published online June 1, 2005
A.R. Medghalchi and T. Yazdanpanah
Teacher Training University, Persian Gulf University
Abstract : 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
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