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

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!

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 $$2, use Kadec-Pelczynski to see that $$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 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 $$1 case, you can improve the constant by using the result that every subspace of $$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.