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spaces of linear and compact operators; non existence of projections; copies of $c_0$; Approximation properties; non existence of norm one projection; Hahn-Banach extensions
In the first part of the paper we prove some new result improving all those already known about the equivalence of the nonexistence of a projection (of any norm) onto the space of compact operators and the containment of $c_0$ in the same space of compact operators. Then we show several results implying that the space of compact operators is uncomplemented by norm one projections in larger spaces of operators. The paper ends with a list of questions naturally rising from old results and the results in the paper.
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