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Uncomplementability of spaces of compact operators in larger spaces of operators. (English) Zbl 0903.46006
Summary: 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.

##### MSC:
 46A32 Spaces of linear operators; topological tensor products; approximation properties 46B20 Geometry and structure of normed linear spaces 46H10 Ideals and subalgebras 46B99 Normed linear spaces and Banach spaces; Banach lattices 46B25 Classical Banach spaces in the general theory
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