Copies of $$c_0$$ and $$\ell _\infty $$ in Banach spaces of $$w^*$$-$$w$$-continuous symmetric multilinear maps

We study copies of the classical sequence spaces $$c_0$$ and $$\ell _\infty $$ in spaces of $$w^*$$ - $$w$$ continuous symmetric n-linear mappings. More precisely, we consider the Banach space $$\mathcal {L}_{w^*s}(^n X^*, Y)$$ of all $$w^*$$ - $$w$$ continuous symmetric n-linear mappings from $$X^*$$ into Y, together with its closed subspace $$\mathcal {K}_{w^*s}(^nX^*,Y)$$ consisting of compact operators. We prove that $$\ell _\infty $$ embeds in $$\mathcal {K}_{w^*s}(^nX^*,Y)$$ if and only if $$\ell _\infty $$ embeds into either X or Y. As an application of this, we prove that if $$c_{0}$$ embeds in $$K_{w^*s}(^nX^*,Y)$$ , then $$\mathcal {K}_{w^*s}(^nX^*,Y)=\mathcal {L}_{w^*s}(^n X^*, Y)$$ if and only if only one of the following statements is true: