Operator Spaces Containing c₀ OR l∞

Let E, F be either Fréchet or complete DF-spaces and let A(E, F) ⊆ B(E, F) be spaces of operators. Under some quite general assumptions we show that: (i) A(E, F) contains a copy of c₀ if and only if it contains a copy of l_∞; (ii) if c₀ ⊆ A(E, F), then A(E, F) is complemented in B(E, F) if and only if A(E, F) = B(E, F); (iii) if E or F has an unconditional basis and A(E, F) ≠ L(E, F), then A(E, F) ⊇ c₀. The above results cover cases of many clssical operator spaces A. We show also that EεF contains l_∞ if and only if E or F contains l_∞.