%0 Journal Article
%T Uniform Convergence to a Left Invariance on Weakly Compact Subsets
%J Sahand Communications in Mathematical Analysis
%I University of Maragheh
%Z 2322-5807
%A Ghaffari, Ali
%A Javadi, Samaneh
%A Tamimi, Ebrahim
%D 2020
%\ 07/01/2020
%V 17
%N 3
%P 81-91
%! Uniform Convergence to a Left Invariance on Weakly Compact Subsets
%K Banach algebra
%K $varphi$-amenability
%K $varphi$-means
%K Weak almost periodic
%K Weak$^*$ topology
%R 10.22130/scma.2019.100548.540
%X LetÂ $left{a_alpharight}_{alphain I}$ be a bounded net in a Banach algebra $A$ and $varphi$ a nonzero multiplicative linear functional on $A$. In this paper, we deal with the problem of when $|aa_alpha-varphi(a)a_alpha|to0$ uniformly for all $a$ in weakly compact subsets of $A$. We show that Banach algebras associated to locally compact groups suchÂ as Segal algebras and $L^1$-algebras are responsive to this concept. It is also shown that $Wap(A)$ has a left invariant $varphi$-mean if and only if there exists a bounded net $left{a_alpharight}_{alphain I}$ in $left{ain A; varphi(a)=1right}$ such that $|aa_alpha-varphi(a)a_alpha|_{Wap(A)}to0$ uniformly for all $a$ in weakly compact subsets of $A$. Other results in this direction are also obtained.
%U http://scma.maragheh.ac.ir/article_40529_52ec68cb33d86279de572c0696818129.pdf