Complemented subspaces of $\mathcal{L}_{p}$-spaces

Question

In 1968, J. Lindenstrauss and A. Pe{\l}czy'{n}ski posed a problem: Is every complemented subspace $X$ of an $\mathcal{L}_{p}$-space ($1\leq p\leq \infty$) either an $\mathcal{L}_{p}$-space or isomorphic to a Hilbert space? If $p=1$ or $\infty$ and $X$ is infinite-dimensional, $X$ can not be a Hilbert space. In 1969, J. Lindenstrauss and H. P. Rosenthal stated that

Theorem. If $Y$ is an $\mathcal{L}_{p,1+\epsilon}$-space for every $\epsilon>0$ and $X$ is a $1$-complemented subspace of $Y$, then $X$ is an $\mathcal{L}_{p,1+\epsilon}$-space for every $\epsilon>0$.

They mentioned that the proof of the theorem can be seen in J. Lindenstrauss and A. Pe{\l}czy'{n}ski's paper. However, I can not see any of its proof.

Question 1. In what reference is there a proof of Theorem ?

Question 2. If $X$ is a $C$-complemented subspace of $Y$ in Theorem, is $X$ an $\mathcal{L}_{p,\lambda}$-space for some $\lambda$ depending only on $C$ ?

Thank you!

Answer 1

I don't know if there is a good reference for Theorem 1. The Lindenstrauss-Pelczynski proof shows that $Y$ is contractively complemented in $L_p(\mu)$ for some measure $\mu$ (at least for $p$ in the reflexive range), and contractively complemented subspaces of $L_p(\mu)$ are isometrically isomorphic to some $L_p(\nu)$. That is in books, or you can look at Beata Randrianantoanina's 2001 paper. These days we would use ultra products to get the L-P part.

In Question 2, you of course have to assume that $X$ is not isomorphic to a Hilbert space. When $2<p<\infty$, use Kadec-Pelczynski to see that $X$ contains a subspace $1+\epsilon$ isomorphic to $\ell_p$. That condition passes to finite codimensional subspaces, so the argument I outlined in your other thread gives what you want. The case $1<p<2$ follows by duality, and for $p=1$ you use the fact that every non reflexive subspace of $L_1(\mu)$ contains a $1+\epsilon$ copy of $\ell_1$ (this is in Wojtaszczyk's book). In the $1<p<2$ case, you can improve the constant by using the result that every subspace of $L_p(\mu)$ that is isomorphic to $\ell_p$ has a smaller subspace that is $1+\epsilon$ isomorphic to $\ell_p$. The argument for the $p=1$ case almost goes over. Schechtman and I outlined the argument in our 2008 paper in J. Eur. Math. Soc. 10, 1105–1119 because we did not know an adequate reference.