Projections in weakly compactly generated Banach spaces and Chang's conjecture

• Autorzy: Koszmider P.
• Języki publikacji: EN

Abstrakty

EN

Classical results on weakly compactly generated (WCG) Banach spaces imply the existence of projectional resolutions of identity (PRI) and the existence of many projections on separable sub-spaces (SCP). We address the questions if these can be the only projections in a nonseparable WCG space, in the sense that there is a PRI [wzór] such that any projection is the sum of an operator in the closure of the linear span of countably many P ∝'s (in the strong operator topology) and a separable range operator. Wark's modification of Shelah's and Steprans' construction provides an unconditional example for λ ω 2. We note that it is impossible for λ > ω 2.

Słowa kluczowe

EN

weakly compactly generated Banach spaces; Chang's conjecture; few operators

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2005
• Tom: Vol. 11, nr 2
• Strony: 187--205
• Opis fizyczny: Bibliogr. 22 poz.

Twórcy

autor

Koszmider P.

• Departamento de Matemática, Universidade de Säo Paulo, Caixa Postal: 66281, Säo Paulo SP, CEP: 05315-970, Brasil,

Bibliografia

• [1] Amir, D., Lindenstrauss, J., The structure of weakly compact sets in Banach spaces, Ann. of Math. (2) 88 (1968), 35-46.
• [2] Argyros, S. A., Lopez-Abad, J., Todorčevic, S., A class of Banach spaces with few non strictly singular operators, J. Funct. Analysis 222(2) (2005), 306-384.
• [3] Baumgartner, J., Shelah, S., Remarks on superatomic Boolean algebras, Ann. Pure Appl. Logic 33 (1987), 109-129.
• [4] Deville, R., Godefroy, G., Zizler, V., Smoothness and Renormings in Banach Spaces, Longman Scientific & Technical, Harlow, 1993.
• [5] Diestel, J., Geometry of Banach Spaces - Selected Topics, Lecture Notes in Math. 485, Springer, New York, 1975.
• [6] Dow, A., Applications of elementary submodels to topology, Topology Proc. 12(1) (1988), 17-73.
• [7] Dunford, N., Schwartz, J., Linear Operators; Part I. General Theory, Interscience Publishers, Inc., New York, Fourth printing, 1967.
• [8] Kanamori, A., The Higher Infinite. Large Cardinals in Set Theory from Their Beginnings, Springer-Verlag, Berlin, 1994.
• [9] Koszmider, P., On the existence of strong chains in P(ω1)/Fin, J. Symbolic Logic 63(3) (1998), 1055-1062.
• [10] Koszmider, P., On strong chains of uncountable functions, Israel J. Math. 118 (2000), 289-315.
• [11] Kunen, K., Set Theory; An Introduction to Independence Proofs, Stud. Logic Found. Math. 102, North Holland Publishing Co., Amsterdam, 1980.
• [12] Pličko, A. On projective resolutions of the identity operator and Marcusevic's bases. Dokl. Akad. Nauk SSRR 25(2) (1982), 386-389.
• [13] Reif, J., A note on Marcusevic's bases in weakly compactly generated Banach spaces, Coram. Math. Univ. Carolin. 15(2) (1974), 335-340.
• [14] Rowbottom, F., Some strong axioms of infinity incompatible with the axiom of constructibility, Ann. Math. Logic 3 (1971), 1-44.
• [15] Singer, I., Bases in Banach Spaces II, Springer-Verlag, Berlin- New York, 1981.
• [16] Shelah, S., A Banach space with few operators, Israel J. Math. 30 (1978), 181-191.
• [17] Shelah, S., Stepräns, J., A Banach space on which there are few operators, Proc. Amer. Math. Soc. 104 (1988), 101-105.
• [18] Todorčevié, S., Remarks on Martin's axiom and the continuum hypothesis, Canad. J. Math. 43 (1991), 832-851.
• [19] Todorčevié, S., Coherent sequences, in „Handbook of Set-Theory”, (to appear).
• [20] Velleman, D., Simplified morasses, J. Symbolic Logic 49(1) (1984), 257-271.
• [21] Wark, H., A nonseparable reflexive Banach space on which there are few operators, J. London Math. Soc. (2) 64(3) (2001), 675-684.
• [22] Zapletal, J., Strongly almost disjoint functions, Israel J. Math. 97 (1997), 101-111.