A common fixed point theorem for a commuting family of weak$^{\ast }$   continuous nonexpansive mappings
  It is shown that if $S$ is a commuting family of weak$^{\ast }$ continuous nonexpansive mappings acting on a weak$^{\ast }$ compact convex subset $C$ of the dual Banach space $E$, then the set of common fixed points of $S$ is a nonempty nonexpansive retract of $C$. This partially solves a long-standing open problem in metric fixed point theory in the case of commutative semigroups. 
S{\l}awomir Borzdy\'nski and Andrzej Wi\'snicki
math.FA