Uniform Convergence to a Left Invariance on Weakly Compact Subsets

Let {aα}α∈I be a bounded net in a Banach algebra A and φ a nonzero multiplicative linear functional on A. In this paper, we deal with the problem of when ∥aaα − φ(a)aα∥ → 0 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 W ap(A) has a left invariant φ-mean if and only if there exists a bounded net {aα}α∈I in {a ∈ A; φ(a) = 1} such that ∥aaα − φ(a)aα∥W ap(A) → 0 uniformly for all a in weakly compact subsets of A. Other results in this direction are also obtained.