Uniform Convergence to a Left Invariance on Weakly Compact Subsets

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\|\to۰ 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^۱-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)=۱\right\} such that \|aa_\alpha-\varphi(a)a_\alpha\|_{Wap(A)}\to۰ uniformly for all a in weakly compact subsets of A. Other results in this direction are also obtained.

Keywords:

Banach algebra , varphi-amenability , varphi-means , Weak almost periodic , Weak^* topology