Common fixed point property through analysis of retractions

In this paper, we study the problem of the existence of common fixed points for a commuting family of nonexpansive mappings. One of our objectives is to provide a simple proof to a more general version of a well-known result due to Bruck which states that if a Banach space has the weak fixed point property, then it has the weak fixed point property for commuting semigroups. As a consequence, we obtain that if a dual Banach space E with a separable predual has the weak* fixed point property and the fixed point property for spheres, then E has the weak* fixed point property for commuting semigroups. This is done through analysis of existence of the so-called retractions.