TY - JOUR ID - 40529 TI - Uniform Convergence to a Left Invariance on Weakly Compact Subsets JO - Sahand Communications in Mathematical Analysis JA - SCMA LA - en SN - 2322-5807 AU - Ghaffari, Ali AU - Javadi, Samaneh AU - Tamimi, Ebrahim AD - Department of Mathematics, Faculty of Science, University of Semnan, P.O.Box 35195-363, Semnan, Iran. AD - Faculty of Engineering- East Guilan, University of Guilan, P. O. Box 44891-63157, Rudsar, Iran. Y1 - 2020 PY - 2020 VL - 17 IS - 3 SP - 81 EP - 91 KW - Banach algebra KW - $varphi$-amenability KW - $varphi$-means KW - Weak almost periodic KW - Weak$^*$ topology DO - 10.22130/scma.2019.100548.540 N2 - Let  $\left\{a_\alpha\right\}_{\alpha\in I}$ 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 such  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 $\left\{a_\alpha\right\}_{\alpha\in I}$ in $\left\{a\in A;\ \varphi(a)=1\right\}$ 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. UR - https://scma.maragheh.ac.ir/article_40529.html L1 - https://scma.maragheh.ac.ir/article_40529_39e3ab77ffd834379f5aaaed481bdb1a.pdf ER -