We treat several questions related to the positions of subspaces of $\ell_1$. Among them, we show that all quotients $\ell_1/\ell_1$ have the Schur property and that a nontrivial twisted sum of $\ell_1$ and $c_0$ cannot be isomorphic to the direct product $\ell_1 \oplus c_0$.
References
A. Avilés, F. Cabello, J.M.F. Castillo, M. González, Y. Moreno, Separably injective Banach spaces. Lecture Notes in Math. 2132, Springer; in press. MR3003929 10.1016/j.aim.2012.10.013 A. Avilés, F. Cabello, J.M.F. Castillo, M. González, Y. Moreno, Separably injective Banach spaces. Lecture Notes in Math. 2132, Springer; in press. MR3003929 10.1016/j.aim.2012.10.013
J. Bourgain and G. Pisier, A construction of ${\mathcal L}_\infty$-spaces and related Banach spaces, Bol. Soc. Bras. Mat. 14 (1983) 109–123. MR756904 10.1007/BF02584862 J. Bourgain and G. Pisier, A construction of ${\mathcal L}_\infty$-spaces and related Banach spaces, Bol. Soc. Bras. Mat. 14 (1983) 109–123. MR756904 10.1007/BF02584862
Y.A. Brudnyi, and N. J. Kalton, Polynomial approximation on convex subsets of $R\sp n$. Constr. Approx. 16 (2000) 161–199. MR1735240 10.1007/s003659910008 Y.A. Brudnyi, and N. J. Kalton, Polynomial approximation on convex subsets of $R\sp n$. Constr. Approx. 16 (2000) 161–199. MR1735240 10.1007/s003659910008
F. Cabello Sánchez, J.M.F. Castillo, N.J. Kalton and D.T. Yost, Twisted sums with $C(K)$ spaces, Trans. Amer. Math. Soc. 355 (2003) 4523–4541. MR1990760 10.1090/S0002-9947-03-03152-0 F. Cabello Sánchez, J.M.F. Castillo, N.J. Kalton and D.T. Yost, Twisted sums with $C(K)$ spaces, Trans. Amer. Math. Soc. 355 (2003) 4523–4541. MR1990760 10.1090/S0002-9947-03-03152-0
J.M.F. Castillo and Y. Moreno, On the Lindenstrauss-Rosenthal theorem, Israel J. Math. 140 (2004) 253–270. MR2054847 10.1007/BF02786635 J.M.F. Castillo and Y. Moreno, On the Lindenstrauss-Rosenthal theorem, Israel J. Math. 140 (2004) 253–270. MR2054847 10.1007/BF02786635
J.M.F. Castillo and A. Plichko, Banach spaces in various positions. J. Funct. Anal. 259 (2010) 2098-2138. MR2671123 10.1016/j.jfa.2010.06.013 J.M.F. Castillo and A. Plichko, Banach spaces in various positions. J. Funct. Anal. 259 (2010) 2098-2138. MR2671123 10.1016/j.jfa.2010.06.013
N.J. Kalton, On subspaces of $c_0$ and extension of operators into $C(K)$-spaces, Q. J. Oxford 52 (2001) 313-328. MR1865904 10.1093/qjmath/52.3.313 N.J. Kalton, On subspaces of $c_0$ and extension of operators into $C(K)$-spaces, Q. J. Oxford 52 (2001) 313-328. MR1865904 10.1093/qjmath/52.3.313
N. Kalton, Automorphisms of $\mathcal{C}(K)$ spaces and extension of linear operators, Illinois J. Math. 52 (2008) 279–317. MR2507245 euclid.ijm/1242414132
N. Kalton, Automorphisms of $\mathcal{C}(K)$ spaces and extension of linear operators, Illinois J. Math. 52 (2008) 279–317. MR2507245 euclid.ijm/1242414132
N.J. Kalton, A. Pełczyński, Kernels of surjections from $\mathcal L_1$-spaces with an application to Sidon sets, Math. Ann. 309 (1997) 135–158. MR1467651 10.1007/s002080050107 N.J. Kalton, A. Pełczyński, Kernels of surjections from $\mathcal L_1$-spaces with an application to Sidon sets, Math. Ann. 309 (1997) 135–158. MR1467651 10.1007/s002080050107
J. Lindenstrauss, A remark on $\mathcal L_1$-spaces Israel J. Math. 8 (1970), 80-82. MR259582 10.1007/BF02771554 J. Lindenstrauss, A remark on $\mathcal L_1$-spaces Israel J. Math. 8 (1970), 80-82. MR259582 10.1007/BF02771554
Y. Moreno, The diagonal functors, Appl. Cat. Struct. 16 (2008) 617–627. MR2448146 10.1007/s10485-007-9074-7 Y. Moreno, The diagonal functors, Appl. Cat. Struct. 16 (2008) 617–627. MR2448146 10.1007/s10485-007-9074-7