Suppose that $X$ is an infinite dimensional subspace of $L^{1}$. In some cases it is true that $X$ contains an isomorphic copy of an infinite dimensional Hilbert space. However, it is not the case when $X$ is a subspace of $\ell_1$ (orthonormal basis converges weakly to zero, but not strongly). I am curious if it is the only obstacle. More precisely: is it true that if an infinite dimensional subspace $X$ of $L^1$ does not contain an infinite dimensional Hilbertian subspace then it embeds in $\ell_1$? If not, do we know anything about such subspaces? I know that the question is a bit vague, but I hope that satisfying answers can be given.
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asked Oct 10, 2014 at 10:43
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$\begingroup$ Are you trying to understand which subspaces of $L_p$ embed into $\ell_p$? They are characterized for $1<p<\infty$ and much is known for $p=1$. $\endgroup$– Bill JohnsonCommented Oct 10, 2014 at 14:50
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$\begingroup$ @BillJohnson, not really, I am trying to investigate non-maximal subspaces of maximal operator spaces, using methods from Banach space theory in this case. $\endgroup$– Mateusz WasilewskiCommented Oct 10, 2014 at 17:35
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$L^1$ contains a copy of $\ell_q$ for every $q\in[1,2]$; I will come back and provide an original reference shortly, however to read about it you probably can't do better than the book Topics in Banach space theory by Albiac and Kalton.
More information in the direction of your question was provided by David Aldous, who showed that every infinite dimensional subspace of $L^1$ contains a subspace isomorphic to $\ell_q$ for some $q\in [1,2]$. Aldous' paper is Subspaces of $L^1$, via random measures, in volume 267 of Transactions of the AMS.
Soon after Aldous' result, Krivine and Maurey proved a more general result, namely that every stable Banach space contains a copy of some $\ell_p$.
Also, I think that David Garling published (in Lecture Notes in Mathematics?) an account of the work of Aldous and of Krivine Maurey; when I get a spare moment I will come back and update the references with additional information.
answered Oct 10, 2014 at 10:52
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$\begingroup$ Ok, thank you very much; I forgot about stable random variables. Of course, $\ell_q$ for $q < 2$ cannot contain a subspace isomorphic to $\ell_2$ by Pitt's theorem. $\endgroup$ Commented Oct 10, 2014 at 12:31