Uncomplemented copies of $\ell_2$ in $L_p(0,1)$ for $2-\varepsilon<p<2$

Question

It was proved in [Bennett, G.; Dor, L.E.; Goodman, V.; Johnson, W.B.; Newman, C.M. On uncomplemented subspaces of $L_p$, $1<p<2$. Israel J. Math. 26 (1977), 178–187] that, for $1 <p < 2$, there is an uncomplemented subspace of $L_p(0,1)$ that is isomorphic to $\ell_2$.

Moreover it is well-known that if $1<p<q<2$ and $M$ is an uncomplemented copy of $\ell_2$ in $L_q(0,1)$, then $M$ is also an uncomplemented copy of $\ell_2$ in $L_p(0,1)$.

My question: Is it possible to find a subspace $M$ of $L_2(0,1)$ such that, for some $\varepsilon>0$, $M$ is an uncomplemented copy of $\ell_2$ in $L_p(0,1)$ for $2-\varepsilon <p<2$ (hence for $1\leq p<2$)?.

Answer 1

Yes. The idea is to find a sequence of finite dimensional subspaces $E_n$ of $L_n$ on which the $L_1$ and $L_2$ norms are uniformly equivalent and such that for all $p<2$ the norm of the best projection from $L_p$ onto $E_n$ tends to infinity as $n\to \infty$, and piece these together to get an infinite dimensional Hilbert space. By Kashin or Figiel, Lindenstraus, and Milman (or a remark in the BDGJN paper you cited) you can take $E_n\subset L_2\{-1,1\}^n$ of proportional dimension $c 2^n$ ($c>0$ independent of $n$) on which $\|\cdot \|_2 \le 2\|\cdot\|_1$, and all the functions in $E_n$ have mean zero. Take an independent sum of these in $L_2\{-1,1\}^{N}$, where $N $ is a disjoint union over $n=1,2,3,\dots$ of $\{1,2,\dots, n\}$ and $L_2\{-1,1\}^n$ is regarded as the functions in $L_2\{-1,1\}^{N}$ that depend only on the $n$th set $\{1,2,\dots, n\}$.