This is an announcement for the paper "Weak$^*$ closures and derived sets in dual Banach spaces" by Mikhail I. Ostrovskii. Abstract: The main results of the paper: {\bf (1)} The dual Banach space $X^*$ contains a linear subspace $A\subset X^*$ such that the set $A^{(1)}$ of all limits of weak$^*$ convergent bounded nets in $A$ is a proper norm-dense subset of $X^*$ if and only if $X$ is a non-quasi-reflexive Banach space containing an infinite-dimensional subspace with separable dual. {\bf (2)} Let $X$ be a non-reflexive Banach space. Then there exists a convex subset $A\subset X^*$ such that $A^{(1)}\neq {\overline{A}\,}^*$ (the latter denotes the weak$^*$ closure of $A$). {\bf (3)} Let $X$ be a quasi-reflexive Banach space and $A\subset X^*$ be an absolutely convex subset. Then $A^{(1)}={\overline{A}\,}^*$. Archive classification: math.FA Mathematics Subject Classification: primary 46B10; secondary 46B15; 46B20 Submitted from: ostrovsm@stjohns.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1003.5176 or http://arXiv.org/abs/1003.5176