I would like to know if there exist an explicit decription of the ultrapowers of $c_0(\ell_1)$ and $\ell_1(c_0)$. The best option would be -- "they are complemented subspaces of $C(K, L_1(\mu))$ and $L_1(\mu, C(K))$ respectively".

It seems to good to be true though, because for $\ell_p(\ell_q)$, where $1\leq p, q<\infty$ the answer is: "they are order ideals of Banach lattices of the form $L_p(\mu, L_q(\nu))$".

I would like to know if there exist an explicit decription of ultrapowers of $c_0(\ell_1)$ and $\ell_1(c_0)$. The best option would be -- "they are complemented subspaces of $C(K, L_1(\mu))$ and $L_1(\mu, C(K))$ respectively".

It seems to good to be true though, because for $\ell_p(\ell_q)$, where $1\leq p, q<\infty$ the answer is: "they are order ideals of Banach lattices of the form $L_p(\mu, L_q(\nu))$".