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Let $p>2$. Following from M.I. Kadec and A. Pełczyński's results (Studia Math. 1962), R.J.Whitley (Trans. Amer. Math.Soc. 1964) observed that $L_{p}$ is subprojective, that is, every infinite-dimensional closed subspace of $L_{p}$ contains an infinite-dimensional closed subspace that is complemented in $L_{p}$. What I am thinking about is the universal subprojectivity of $L_{p}$. Namely, I have the following question:

Question: Does every infinite-dimensional closed subspace of $L_{p}$ contain an infinite-dimensional closed subspace that is $C_{p}$-complemented in $L_{p}$? where the constant $C_{p}$ depends only on $p$.

Let $E$ be an infinite-dimensional closed subspace of $L_{p}$. If $E$ is not isomorphic to $l_{2}$, then an observation of the proof of Theorem 2 in M.I. Kadec and A. Pełczyński's paper (Studia Math. 1962) shows that $E$ contains an infinite-dimensional closed subspace that is $\lambda_{p}$-complemented in $L_{p}$, where the constant $\lambda_{p}$ depends only on $p$. But, I do not know whether this is true for the case $E$ is isomorphic to $l_{2}$.

Thank you!

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Since the space $L_p$ $(1\le p<\infty)$ is stable in the sense of Krivine and Maurey (Espaces de Banach stables, Israel J. Math. 39 (1981), no. 4, 273–295), you can combine their result in this paper and get that any subspace isomorphic to $\ell_2$ in $L_p$ contains a subspace which is $(1+\varepsilon)$-isometric to $\ell_2$. Then you apply the result of Maurey (Un théorème de prolongement. (French) C. R. Acad. Sci. Paris Sér. A 279 (1974), 329–332), and get the desired estimate if $p>2$.

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  • $\begingroup$ Theorem 1.3 in D.Alspach's paper(Banach J.Math.Anal.2009) shows that any subspace isomorphic to $l_{2}$ in $L_{p}(p>2)$ contains a subspace which is $(1+\epsilon)$-isometric to $l_{2}$ and is $(1+\epsilon)\gamma_{p}$-complemented in $L_{p}$, where $\gamma_{p}$ is the norm of a symmetric Gaussian random variable. Since I do not download Maurey's paper, I do not know the desired estimate you said is the same as $(1+\epsilon)\gamma_{p}$. Thank you. $\endgroup$ – Dongyang Chen Jun 1 '16 at 18:11
  • $\begingroup$ I do not have the paper of Maurey either, as far as I remember the constant is the type 2 constant of $L_p$. $\endgroup$ – Mikhail Ostrovskii Jun 1 '16 at 18:19

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