Every complemented subspace of a $C(T)$-space is an $\mathcal{L}_{\infty}$-space

Question

Let $T$ be a compact Hausdorff space and $X$ be an infinite-dimensional complemented subspace of $C(T)$.

Question 1. Assume that $X$ has a subspace $U$ that is isomorphic to $c_{0}$. Given any positive integer $n$ and a finite-dimensional subspace $M$ of $X$. Do there exist a constant $K$, depending only on the Banach-Mazur distance $\textrm{d}(U,c_{0})$, and a $n$-dimensional subspace $N$ of $U$ so that the Banach-Mazur distance $\textrm{d}(N,l_{\infty}^{n})\leq K$ and $$\max(\|x\|,\|y\|)\leq K\|x+y\|, \quad x\in M,y\in N ?$$

Question 2. J. Lindenstrauss and H. P. Rosenthal (1969) proved that $X$ is an $\mathcal{L}_{\infty,\lambda}$-space for some $\lambda$. But I want to know more about it. It is known that $C(T)$ is an $\mathcal{L}_{\infty,1+\epsilon}$ for every $\epsilon>0$. We further assume that $X$ is $C$-complemented in $C(T)$. Then what is the $\lambda$ ?

Thank you!

Answer 1

Question 1 has an affirmative answer. You don't need $X$ complemented in a $C(K)$ space and for $U$ you only need that it fails cotype. Use the Mazur technique for constructing basic sequences: norm to $1+\epsilon$ the finite dimensional $M$ by finitely many linear functionals of norm one. The intersection of the kernels of these functionals with $U$ (call the intersection $U_1$) is a finite codimensional subspace of $U$, and hence $U_1$ also fails cotype, which implies that $U_1$ contains for all $n$ a $1+\epsilon$ isomorphic copy of $\ell_\infty^n$. This gives you what you want with $K$ at most $2+\epsilon$.

I think that you know everything I said, so maybe I misunderstood your question?