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...

The concept of uniform convexity of a Banach space was gen- eralized to linear operators between Banach spaces and studied by Beauzamy [1]. Under this generalization, a Banach space X is uniformly convex if and only if its identity map Ix is. Pisier showe

We give counterexamples to a conjecture of Bourgain, Casazza, Lindenstrauss and Tzafriri that if X has a unique unconditional basis (up to permutation) then so does $c_0\left(X\right)$. We also give some positive results including a simpler proof that $c_0\left({\ell }_1\right)$ has a unique unconditional basis and a proof that $c_0\left({\ell }_{p_n}^{N_n}\right)$ has a unique unconditional basis when $p_n￬1$, $N_{n+1}\ge 2N_n$ and $(p_n-p_{n+1})log{N}_n$ remains bounded.