ON THE BEST STABILITY BOUNDS FOR A WEAK STAR FIXED POINT PROPERTY

It is well-known that the Banach space (l(1), parallel to center dot parallel to ) satisfies the w*-FPP when l(1) is considered the dual of co. This property is inherited by all the equivalent norms on l(1) whose Banach-Mazur distance with the parallel to center dot parallel to(1) norm is strictly less than 2. This is the best w*-FPP stability constant that can be achieved for the parallel to center dot parallel to(1) norm, since l(1) can be renormed in such a way that (l(1), vertical bar center dot vertical bar) fails to have the w*-FPP and the Banach Mazur distance between (l(1), parallel to center dot parallel to(1)) and (l(1), vertical bar center dot vertical bar) is equal to 2. We will prove that this is a general statement for every reforming of l(1). That is, for every equivalent norm vertical bar center dot vertical bar(1) and for every epsilon > 0, there exists vertical bar center dot vertical bar(2) such that (l(1), vertical bar center dot vertical bar(2)) fails to have the w*-FPP and the Banach Mazur distance between both norms is less than 2 + epsilon. As a conclusion, 2 is the best possible stability constant for the w*-FPP for every equivalent norm on l(1). We will extend this result to more general dual Banach spaces as well as to non-dual Banach spaces endowed with different topologies.