The FPP for left reversible semigroups in ℓ1 endowed with different locally convex topologies

Abstract

In this article we will consider locally convex topologies $\tau$ on ${\ell }_1$ which are coarser than the weak topology on the unit ball and such that the unit vector basic sequence $\left(e_n\right)$ is $\tau$-convergent. We characterize these topologies depending on the $\tau$-fixed point property for left reversible semigroups on $({\ell }_1,\Vert \cdot {\Vert }_1)$. We will apply our results to the case of different weak${}^{\ast }$ topologies on ${\ell }_1$.