1
$\begingroup$

In 1974, W. B. Johnson and E. Odell observed that there are subspaces $X$ of $L_{1}$ with the Schur property. In 1980, J. Bourgain and H. P. Rosenthal constructed a subspace $X$ of $L_{1}$ such that $X$ has the Schur property, but $X$ is not isomorphic to a subspace of $l_{1}$. Hence, I have the first question as follows:

Question 1. Is every Banach space with the Schur property isomorphic to a subspace of $L_{1}(\mu)$ for some measure $\mu$?

Moreover, W. B. Johnson and E. Odell gave natural non-trivial conditions that a subspace of $L_{p}$ embeds into $l_{p}$ for $1<p<\infty, p\neq 2$. But

Question 2. Are there conditions that a subspace of $L_{1}$ embeds into $l_{1}$ ?

Thank you!

Improve this question
$\endgroup$

1 Answer 1

Reset to default
3
$\begingroup$

Absolutely not. Take the the $\ell_1$-sum of $\ell_\infty^n$ ($n\in \mathbb N$). If that embedded into $L_1(\mu)$, then you would have found $c_0$ in some ultrapower of $L_1(\mu)$, which is impossible.

Improve this answer
$\endgroup$
2
  • $\begingroup$ why is $c_{0}$ in some ultrapower of $L_{1}(\mu)$? $\endgroup$
    – Dongyang Chen
    Commented Mar 22, 2018 at 15:31
  • 1
    $\begingroup$ @DongyangChen, that is the standard trick linking finite-representability and embeddability into ultrapowers. $\endgroup$
    – Tomasz Kania
    Commented Mar 22, 2018 at 15:40

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .