It is well known that neither
1) $c_0$ is isomorphic to a complemented subspace of $\mathcal{B}(H)$
nor
2) $c_0$ is a quotient of $\mathcal{B}(H)$
for a Hilbert space $H$. Can we replace $H$ above by any Banach space? Reflexive space?
It is well known that neither 1) $c_0$ is isomorphic to a complemented subspace of $\mathcal{B}(H)$ nor 2) $c_0$ is a quotient of $\mathcal{B}(H)$ for a Hilbert space $H$. Can we replace $H$ above by any Banach space? Reflexive space? 


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I am pretty sure that this is still open, though there are many partial results. Look up, e.g., papers by Giovanni Emmanuele on MathSciNet. 


$c_0$ is a quotient of ${\cal B}(c_0)$, namely for any nonzero $x \in c_0$, the evaluation map $T \to T(x)$ maps ${\cal B}(c_0)$ onto $c_0$. 

