$X^*$ with weak* uniform Kadec-Klee property has Property($K^*$)

Abstract

It is shown that if the dual of a Banach space, $X^*$, where the dual ball is weak* sequentially compact, has the weak* uniform Kadec-Klee property then $X^*$ has Property($K^*$). An example is given where the reverse implication does not hold. That is, there is a Banach space $X$ whose dual, $X^*$, has Property($K^*$) but $X^*$ does not have the weak* uniform Kadec-Klee property.