Banach limits and uniform almost summability

Some classical Hahn-Schur Theorem-like results on the uniform convergence of unconditionally convergent series can be generalized to weakly unconditionally Cauchy series. In this paper, we obtain this type of generalization via a summability method based upon the concept of almost convergence. We also obtain a generalization of the main result in Aizpuru et al. (2003) [3] using pointwise convergence of sums indexed in natural Boolean algebras with the Vitali-Hahn-Saks Property. In order to achieve that, we first study the notion of almost convergence through its original definition (which involves Banach limits), giving a description of the extremal structure of the set of all norm-1 Hahn-Banach extensions of the limit function on c to ℓ∞. We also show the existence of norm-1 Hahn-Banach extensions of the limit function on c to ℓ∞ that are not extensions of the almost limit function and hence are not Banach limits.