Description:
<jats:p>In this paper the result of Sobczyk about complemented copies of <jats:italic>c</jats:italic><jats:sub>0</jats:sub> is extended to a class of Banach spaces <jats:italic>X</jats:italic> such that the unit ball of their dual endowed with the weak* topology has a certain topological property satisfied by every Corson-compact space. By means of a simple example it is shown that if Corson-compact is replaced by Rosenthal-compact, this extension does not hold. This example gives an easy proof of a result of Phillips and an easy solution to a question of Sobczyk about the existence of a Banach space <jats:italic>E</jats:italic>, <jats:italic>c</jats:italic><jats:sub>0</jats:sub> ⊂ <jats:italic>E</jats:italic> ⊂ <jats:italic>l</jats:italic>∞, such that <jats:italic>E</jats:italic> is not complemented in <jats:italic>l</jats:italic>∞ and <jats:italic>c</jats:italic><jats:sub>0</jats:sub> is not complemented in <jats:italic>E</jats:italic>. Assuming the continuum hypothesis, it is proved that there exists a Rosenthal-compact space <jats:italic>K</jats:italic> such that <jats:italic>C</jats:italic>(<jats:italic>K</jats:italic>) has no projectional resolution of the identity.</jats:p>