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

TL;DR

The paper proves that for dual Banach spaces with weak* sequential compactness, the weak* uniform Kadec-Klee property implies Property(K*), and provides a counterexample for the reverse implication.

Contribution

It establishes a one-way implication between the weak* uniform Kadec-Klee property and Property(K*) in dual Banach spaces under certain conditions, and presents a counterexample.

Findings

Weak* uniform Kadec-Klee property implies Property(K*) in dual Banach spaces.

Counterexample shows the reverse implication does not hold.

Dual spaces with weak* sequential compactness are key to the result.

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.