Let $X$ be a Banach space. If we know that $X^*$ has a complemented subspace isomorphic to $\ell_2$, then can we say that $X$ has a complemented subspace isomorphic to $\ell_2$? Precisely, I want to know whether the predual of a type III von Neumann algebra has an $\ell_2$-complemented copy.
2026-03-30 06:46:58.1774853218
does $X$ has an $\ell_2$-complemented copy if the dual $X^*$ has an $\ell_2$-complemented copy
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