On asymptotic structure, the Szlenk index and UKK properties in Banach spaces

Let B be a separable Banach space and let X = B* be separable. We prove that if B has finite Szlenk index (for all epsilon > 0) then B can be renormed to have the weak* uniform Kadec-Klee property. Thus if epsilon > 0 there exists delta (epsilon) > 0 so that if (x(n)) is a sequence in the ball of X converging omega* to x so that lim inf(n -->infinity) parallel to x(n)-x parallel to greater than or equal to epsilon then parallel to x parallel to less than or equal to 1-delta (epsilon). In addition we show that the norm can be chosen so that delta (epsilon) greater than or equal to c epsilon(p) for some p < infinity and c > 0.