Copies of c₀ and _ in Banach spaces of w^*-w-continuous symmetric multilinear maps

We study copies of the classical sequence spaces c₀ and _ in spaces of w^* - w continuous symmetric n-linear mappings. More precisely, we consider the Banach space Lₖ^*ₒ (ⁿ X^*, Y) of all w^* - w continuous symmetric n-linear mappings from X^* into Y, together with its closed subspace Kₖ^*ₒ (ⁿX^*, Y) consisting of compact operators. We prove that _ embeds in Kₖ^*ₒ (ⁿX^*, Y) if and only if _ embeds into either X or Y. As an application of this, we prove that if c₀ embeds in Kₖ^*ₒ (ⁿX^*, Y), then Kₖ^*ₒ (ⁿX^*, Y) = Lₖ^*ₒ (ⁿ X^*, Y) if and only if only one of the following statements is true: