An isomorphic classification of C(2m x [0, α]) spaces

• Autorzy: Galego E. M.
• Języki publikacji: EN

Abstrakty

EN

We present an extension of the classical isomorphic classification of the Banach spaces C([0, α]) of all real continuous functions defined on the nondenumerable intervals of ordinals [0, α]. As an application, we establish the isomorphic classification of the Banach spaces C(2m x [0, α]) of all real continuous functions defined on the compact spaces 2m x [0, α], the topological product of the Cantor cubes 2m with m smaller than the first sequential cardinal, and intervals of ordinal numbers [0, α]. Consequently, it is relatively consistent with ZFC that this yields a complete isomorphic classification of C(2m x [0, α]) spaces.

Słowa kluczowe

EN

Banach spaces of continuous functions; Cantor cube; isomorphic classification; Mazur property; sequential cardinal

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2009
• Tom: Vol. 57, no 3-4
• Strony: 279--287
• Opis fizyczny: Bibliogr. 28 poz.

Twórcy

autor

Galego E. M.

• Department of Mathematics, University of Sao Paulo, Sao Paulo, Brazil 05508-090,

Bibliografia

• [1] M. Ya. Antonovskiĭ and D. V. Chudnovskiĭ, Some questions of general topology and Tikhonov semifields II, Russian Math. Surveys 31 (1976), 69-128.
• [2] C. Bessaga and A. Pełczyński, Spaces of continuous functions IV, Studia Math. 19 (1960), 53-61.
• [3] L. Burlando, On subspaces of direct sums of infinite sequences of Banach spaces, Atti Acad. Ligure Sci. Lett. 46 (1990), 96-105.
• [4] G. A. Edgar, Measurability in a Banach space II, Indiana Univ. Math. J. 28 (1977), 559-579.
• [5] R. Engelking, General Topology, Sigma Ser. Pure Math. 6, Heldermann, Berlin, 1989.
• [6] E. M. Galego, How to generate new Banach spaces non-isomorphic to their Cartesian squares, Bull. Polish Acad. Sci. Math. 47 (1999), 1, 21-25.
• [7] E. M. Galego, Banach spaces of continuous vector-valued functions of ordinals, Proc. Edinburgh Math. Soc. (2) 44 (2001), 49-62.
• [8] E. M. Galego, On isomorphism classes of C(2m○[0, α]) spaces, Fund. Math. 204 (2009), 87-95.
• [9] E. M. Galego, Complete isomorphic classification of some spaces of compact operators, Proc. Amer. Math. Soc. 138 (2010), 725-736.
• [10] S. P. Gul’ko and A. V. Os’kin, Isomorphic classification of spaces of continuous functions on totally ordered bicompacta, Functional Anal. Appl. 9 (1975), 56-57.
• [11] T. Jech, Set Theory, Academic Press, New York, 1978.
• [12] W. B. Johnson and J. Lindenstrauss, Basic concepts in the geometry of Banach spaces, in: Handbook of the Geometry of Banach Spaces, North-Holland, Amsterdam, 2001, 1-84.
• [13] T. Kappeler, Banach spaces with the condition of Mazur, Math. Z. 191 (1986), 623-631.
• [14] S. V. Kislyakov, Classification of spaces of continuous functions of ordinals, Siberian Math. J. 16 (1975), 226-231.
• [15] M. A. Labbé, Isomorphisms of continuous function spaces, Studia Math. 52 (1975), 221-231.
• [16] D. Leung, Banach spaces with Mazur’s property, Glasgow Math. J. 33 (1991), 51-54.
• [17] S. Mazur, On continuous mappings on Cartesian products, Fund. Math. 39 (1952), 229-238.
• [18] S. Mazurkiewicz et W. Sierpiński, Contribution à la topologie des ensembles dénombrables, Fund. Math. 1 (1920), 17-27.
• [19] N. Noble, The continuity of functions on Cartesian products. Trans. Amer. Math. Soc. 149 (1970), 187-198.
• [20] G. Plebanek, On Pettis integrals with separable range, Colloq. Math. 64 (1993), 71-78.
• [21] G. Plebanek, On some properties of Banach spaces of continuous functions, in: Seminaire d’Initiation à l’Analyse, exp. 20, Publ. Math. Univ. Pierre et Marie Curie 107, Univ. Paris VI, Paris, 1991/1992, 9 pp.
• [22] H. P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory, Studia Math. 37 (1970), 13-36.
• [23] H. P. Rosenthal, On injective Banach spaces and the spaces L∞ (μ) for finite measures μ, Acta Math. 124 (1970), 205-248.
• [24] H. P. Rosenthal, On quasi-complemented subspaces of Banach spaces, with an appendix on compactness of operators from Lp(μ) to Lr(ν), J. Funct. Anal. 4 (1969), 176-214.
• [25] C. Samuel, On spaces of operators on C(Q) spaces (Q countable metric space), Proc. Amer. Math. Soc. 137 (2009), 965-970.
• [26] Z. Semadeni, Banach spaces non-isomorphic to their Cartesian squares. II, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 8 (1960), 81-84.
• [27] Z. Semadeni, Banach Spaces of Continuous Functions, Vol. I, Monografie Mat. 55, PWN-Polish Sci. Publ., Warszawa, 1971.
• [28] A. Wilansky, Mazur spaces, Int. J. Math. Math. Sci. 4 (1981), 39-53.