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

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!

• @TomaszKania I just want to estimate how many the $\lambda$ is, depending only on the constant $C$, not to say that $X$ is an $\mathcal{L}_{\infty,1+}$. That is, I want to prove a quantitative version of Theorem 3.2 in J. Lindenstrauss and H. P. Rosenthal's paper in 1969. Apr 13 at 13:24
• Question 2 follows from Question 1. Apr 13 at 14:35

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 $$2+\epsilon$$.
• As for Question 2, J. Lindenstrauss and H. P. Rosenthal pointed out that it follows from the proof of Theorem 2.1 and James's distortion theorem that if $X$ is $C$-complemented in $C(K)$ , then $X$ is an $\mathcal{L}_{\infty, 9C+\epsilon}$ space for every $\epsilon>0$. Apr 21 at 15:28
• In you answer, norm to $1+\epsilon$ the finite dimensional $M$ by finitely many linear functionals of norm one may means that there exists $x^{*}_{1},\cdots,x^{*}_{n}$ of norm one in $X^{*}$ so that $\max_{k}|\langle x^{*}_{k},m\rangle\|\geq \frac{1}{1+\epsilon}\|m\|$ for $m\in M$. Is that right? Apr 22 at 14:01