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Complementation and decompositions in some weakly Lindelöf Banach spaces. (English) Zbl 1225.46012
Consider the class \(\mathcal{L}\) of all scattered compact spaces \(K\) of countable height such that \(C(K)\) is Lindelöf. The authors show that the following statement is independent of ZFC:
“For every \(K\in \mathcal{L}\) and for every copy of \(c_0(\omega_1)\) as a subspace of \(C(K)\) there exists an uncountable subset \(E\subset\omega_1\) such that \(c_0(E)\) is a complemented subspace of \(C(K)\)”.
This fact was known to be true in ZFC under the additional assumption that \(K\) is an Eberlein compact [S. A. Argyros, J. F. Castillo, A. S. Granero, M. Jiménez, and J. P. Moreno, Proc. Lond. Math. Soc., III. Ser. 85, No. 3, 742–768 (2002; Zbl 1017.46011)]. Now, the authors prove it to be true under the P-ideal dichotomy, but false under \(\clubsuit\). Indeed, a space \(K\in\mathcal{L}\) is constructed under \(\clubsuit\) such that in any decomposition \(C(K) = A\oplus B\) into infinite-dimensional spaces, one of the factors is isomorphic to \(C(K)\) and the other one is isomorphic to either \(c_0\) or \(C(\omega^\omega)\) (moreover, every operator \(T:C(K)\to C(K)\) is a multiple of the identity plus an operator with separable range).
The paper also includes some general facts about compact spaces \(K\in\mathcal{L}\) and their \(C(K)\) spaces. The authors show that, when \(K\in \mathcal{L}\) is uncountable and has finite height, then \(C(K)\) always contains complemented copies of \(c_0(\omega_1)\), but – unlike the case of Eberlein compacta – there exists such a space with also an uncomplemented copy of \(c_0(\omega_1)\). They pose the question whether the class \(\mathcal{L}\) coincides with the class of compact scattered spaces of countable height in which the closure of every countable space is countable.

MSC:
46B26 Nonseparable Banach spaces
54C35 Function spaces in general topology
03E35 Consistency and independence results
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
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[1] Amir, D.; Lindenstrauss, J., The structure of weakly compact sets in Banach spaces, Ann. of math. (2), 88, 35-46, (1968) · Zbl 0164.14903
[2] Argyros, S.; Castillo, J.; Granero, A.; Jiménez, M.; Moreno, J., Complementation and embeddings of \(c_0(I)\) in Banach spaces, Proc. lond. math. soc. (3), 85, 3, 742-768, (2002) · Zbl 1017.46011
[3] Argyros, S.; Lopez-Abad, J.; Todorcevic, S., A class of Banach spaces with few non-strictly singular operators, J. funct. anal., 222, 306-384, (2005) · Zbl 1086.46005
[4] Argyros, S.; Raikoftsalis, T., Banach spaces with a unique nontrivial decomposition, Proc. amer. math. soc., 136, 10, 3611-3620, (2008) · Zbl 1162.46011
[5] Arhangelski, A., Topological function spaces, Math. appl., vol. 78, (1992), Kluwer Academic Publishers Dordrecht
[6] Bessaga, Cz.; Pełczyński, A., Spaces of continuous functions (IV) (on isomorphical classification of spaces of continuous functions), Studia math., 19, 53-62, (1960) · Zbl 0094.30303
[7] Engelking, R., General topology, Sigma ser. pure math., vol. 6, (1989), Heldermann Verlag Berlin · Zbl 0684.54001
[8] Fabian, M., Functional analysis and infinite-dimensional geometry, CMS books math./ouvrages math. SMC, vol. 8, (2001), Springer-Verlag New York
[9] Godefroy, G.; Kalton, N.; Lancien, G., Subspaces of \(c_0(\mathbb{N})\) and Lipschitz isomorphisms, Geom. funct. anal., 10, 798-820, (2000) · Zbl 0974.46023
[10] Koppelberg, S.; Rubin, M., A superatomic Boolean algebra with few automorphisms, Arch. math. logic, 40, 125-129, (2001) · Zbl 0977.06006
[11] Koszmider, P., Projections in weakly compactly generated Banach spaces and Chang’s conjecture, J. appl. anal., 11, 2, 187-205, (2005) · Zbl 1101.46013
[12] Koszmider, P., On decompositions of Banach spaces of continuous functions on mrówka’s spaces, Proc. amer. math. soc., 133, 7, 2137-2146, (2005) · Zbl 1085.46015
[13] Kunen, K., Set theory. an introduction to independence proofs, Stud. logic found. math., vol. 102, (1980), North-Holland Amsterdam · Zbl 0443.03021
[14] Marciszewski, W., On Banach spaces \(C(K)\) isomorphic to \(c_0(\Gamma)\), Studia math., 156, 3, 295-302, (2003) · Zbl 1026.46005
[15] Ostaszewski, K., On countably compact, perfectly normal spaces, J. lond. math. soc. (2), 14, 505-516, (1976) · Zbl 0348.54014
[16] Pełczyński, A.; Semadeni, Z., Spaces of continuous functions (III) (spaces \(C(\Omega)\) for ω without perfect sets), Studia math., 18, 211-222, (1959) · Zbl 0091.27803
[17] Pol, R., Concerning function spaces on separable compact spaces, Bull. acad. polon. sci. ser. sci. math. astronom. phys., 25, 10, 993-997, (1977) · Zbl 0389.54009
[18] Pol, R., A function space \(C(X)\) which is weakly Lindelöf but not weakly compactly generated, Studia math., 64, 3, 279-285, (1979) · Zbl 0424.46011
[19] Roitman, J., Basic S and L, (), 295-326
[20] Rosenthal, H., The Banach space \(C(K)\), (), Ch. 36, pp. 1547-1602 · Zbl 1040.46018
[21] Shelah, S., A Banach space with few operators, Israel J. math., 30, 181-191, (1978) · Zbl 0394.46015
[22] Shelah, S., Whitehead groups may not be free, even assuming CH. II, Israel J. math., 35, 257-285, (1980) · Zbl 0467.03049
[23] Shelah, S.; Steprans, J., A Banach space on which there are few operators, Proc. amer. math. soc., 104, 1, 101-105, (1988) · Zbl 0681.46024
[24] Todorcevic, S., Partitioning pairs of countable ordinals, Acta math., 159, 261-294, (1987) · Zbl 0658.03028
[25] Todorcevic, S., A dichotomy for P-ideals of countable sets, Fund. math., 166, 3, 251-267, (2000) · Zbl 0968.03049
[26] Wark, H., A non-separable reflexive Banach space on which there are few operators, J. lond. math. soc. (2), 64, 3, 675-689, (2001) · Zbl 1030.46015
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