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 Lw^*s(ⁿ X^*, Y) of all w^* - w continuous symmetric n-linear mappings from X^* into Y, together with its closed subspace Kw^*s(ⁿX^*,Y) consisting of compact operators. We prove that _∞ embeds in Kw^*s(ⁿX^*,Y) if and only if _∞ embeds into either X or Y. As an application of this, we prove that if c₀ embeds in Kw^*s(ⁿX^*,Y) , then Kw^*s(ⁿX^*,Y)=Lw^*s(ⁿ X^*, Y) if and only if only one of the following statements is true: