# Complemented subspaces of ultrapowers

It's a famous result of Maurey that a Banach space $E$ is finitely representable in $X$ if and only if it is a subspace of some ultrapower of $X$. Is there an analogous result for complemented subspaces? More specifically,

Let $E$ be a Banach space with an FDD $(E_n)$. Suppose that $E$ is isomorphic to a complemented subspace of some ultrapower $X^U$. Are the subspaces $(E_n)$ uniformly complemented in $X$?

EDIT: Here is how to apply Heinrich's Theorem 7.3. It is an elementary observation that an $n$ dimensional subspace $E$ of a Banach space $Y$ is $C$-complemented iff there is an $n$ dimensional subspace $F$ of $Y^*$ that $C$-norms $E$. Suppose that $E \subset X^U$ is $n$ dimensional and $F \subset (X^U)^*$ is $n$ dimensional and $C$-norms $E$. If $F\subset (X^*)^U$, there is no problem pulling back to $X$, and Heinrich's local reflexivity principle for ultrapowers says that you can take $F\subset (X^*)^U$ (which is strictly smaller than $(X^U)^*$ when $X$ is not superreflexive).