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

Abstract

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