Uniform Convergence to a Left Invariance on Weakly Compact Subsets

Let  $left{a_alpharight}_{alphain I}$ 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 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 $left{a_alpharight}_{alphain I}$ 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.