The Space of Continuous Linear Functionals on ℓ1 Approximated by Weakly Symmetric Continuous Linear Functionals

We study the space of continuous linear functionals approximated by weakly symmetric continuous linear functionals on the complex Banach space ℓ1 of all absolutely summing complex sequences. We construct the sequence of groups of symmetries on ℓ1, obtain the structure of corresponding weakly symmetric continuous linear functionals, find the completion of the space, and construct for it a Schauder basis.