The following question is bothering me. Suppose that we have Banach space $X$ such that $X^*$ has a quotient isomorphic to $c_0$. Must $X^*$ contain a complemented copy of $\ell_1$?
Do dual Banach spaces admitting $c_0$ as a quotient contain complemented copies of $\ell_1$?
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functional-analysis
banach-spaces
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1Did you mean "Must $X$ contain a complemented copy of $\ell_1$?" – 2012-06-19
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0@Leonid: I suspect that this question is not so much about duality, but more about universality (in particular, universal surjectivity). Perhaps I am reading too much into it, but the way I interpret the question is: Suppose $X^\ast$ has a quotient isomorphic to $c_0$. Is $X^\ast$ surjectively universal for the class of separable Banach spaces? That is, does $X^\ast$ necessarily admit continuous linear surjections onto *all* separable Banach spaces? Equivalently, does $X^\ast$ contain a complemented subspace isomorphic to $\ell_1$? – 2012-06-21
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0In any case, the answer is *no* for both the OP's question as given and to your suggested possible correction to the question. In the case suggested by you - i.e., *does $X$ contain a complemented copy of $\ell_1$?* - I am sure you have no trouble coming up with a counterexample, e.g., take $X=c_0$ or, more generally, $X=C(K)$ for infinite compact Hausdorff $K$. I will post a negative answer to the OP's question later today when I have the time to do so. – 2012-06-21
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0@PhilipBrooker Thank you. To tell the truth, I did not feel inclined to think about the question without being certain that it's stated correctly. (For one thing, the title says "Dual without quotients isomorphic to $c_0$, contrary to the body of the question.) – 2012-06-21
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0@LeonidKovalev: good point about the title; I actually didn't notice that myself! – 2012-06-21