Abstract.
We present two constructions of infinite, separable, compact Hausdorff spaces K for which the Banach space C(K) of all continuous real-valued functions with the supremum norm has remarkable properties. In the first construction K is zero-dimensional and C(K) is non-isomorphic to any of its proper subspaces nor any of its proper quotients. In particular, it is an example of a C(K) space where the hyperplanes, one co-dimensional subspaces of C(K), are not isomorphic to C(K). In the second construction K is connected and C(K) is indecomposable which implies that it is not isomorphic to any C(K’) for K’ zero-dimensional. All these properties follow from the fact that there are few operators on our C(K)’s. If we assume the continuum hypothesis the spaces have few operators in the sense that every linear bounded operator T : C (K) → C (K) is of the form gI+S where g∈C(K) and S is weakly compact or equivalently (in C(K) spaces) strictly singular.
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References
Argyros, S., Lopez-Abad, J., Todorcevic, S.: A class of Banach spaces with no unconditional basic sequences. Note aux C. R. A. S. Paris 337, 1 (2003)
Arhangel’skii, A. V.: Problems in C p -theory; in Open problems in Topology. J. van Mill, G. M. Reed eds. North Holland 1990
Banach, S.: Théorie des opérations linéaires. Monografje Matematyczne, Państwowe Wydawnictwo Naukowe, 1932
Bessaga, Cz., Pełczyński, A.: Spaces of continuous functions (VI) (On isomorphical classification of spaces C(S)). Studia Math. 19, 53–62 (1960)
Comfort, W., Negrepontis, S.: Chain conditions in topology; Cambridge University Press 1982
Diestel, J.: Sequences and series in Banach spaces; Springer-Verlag 1984
Diestel, J., Uhl Jr, J.J.: Vector Measures; Mathematical Surveys 15, AMS. 1977
Dunford, N., Schwartz, J.: Linear Operators; Part I, General Theory. Interscience Publishers, INC., New York, Fourth printing, 1967
Engelking, R.: General Topology; PWN 1977
Fedorchuk, V. V.: On the cardinality of hereditarily separable compact Hausdorff spaces. Soviet Math. Dokl. 16, 651–655 (1975)
Godefroy, G.: Banach spaces of continuous functions on compact spaces; Ch. 7 in in Recent Progress in General Topology II; eds M. Husek, J. van Mill; Elsevier 2002
Gowers, W. T.: A solution to Banach’s hyperplane problem. Bull. London Math. Soc. 26, 523–530 (1994)
Gowers, W. T., Maurey, B.: The unconditional basic sequence problem. Journal A. M. S. 6, 851–874 (1993)
Haydon, R.: A non-reflexive Grothendieck space that does not contain l∞; Israel J. Math. 40, 65–73 (1981)
Jech, T.: Set Theory. Second edition. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1997
Jimenez, M., Moreno, J.: Renorming Banach spaces with the Mazur intersection property. Jornal of Funct. Anal. 144, 486–804 (1997)
Koszmider, P.: Forcing minimal extensions of Boolean algebras. Trans. Amer. Math. Soc. 351, 3073–3117 (1999)
Koszmider, P.: On decompositions of Banach spaces of continuous functions on Mrówka’s spaces; Preprint 2003
Kunen, K.: Set Theory. An Introduction to Independence Proofs. North Holland, 1980
Lacey, H. E.: Isometric Theory of Classical Banach Spaces: Springer-Verlag 1974
Lacey, E., Morris, P.: On spaces of the type A(K) and their duals. Proc. Amer. Math. Soc. 23, 151–157 (1969)
Lindenstrauss, J.: Decomposition of Banach spaces; Proceedings of an International Symposium on Operator Theory (Indiana Univ., Bloomington, Ind., 1970). Indiana Univ. Math. J. 20 917–919 (1971)
Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces I: Sequence Spaces: Springer Verlag 1977
Maurey, B.: Banach spaces with few operators; Handbook of Geometry of Banach Spaces, Vol 2. Ch. 29, pp. 1247–1297 (eds. W.B. Johnson, J. Lindenstrauss); North Holland 2003.
Mibu: On Baire functions on infinite product spaces. Proc. Imperial Acad. Tokyo (20), (1944)
Marciszewski, W.: A function space C p (X) not linearly homeomorphic to C p (X)× R. Fund. Math. 153, 125–140 (1997)
Miljutin, A. A.: On spaces of continuous functions; Dissertation, Moscow State University, 1952
Negrepontis, S.: Banach spaces and topology; in Handbook of Set-theoretic topology; eds. K Kunen, J Vaughan. North-Holland 1980, pp. 1045–1142
Pełczyński, A.: On strictly singular and strictly cosingular operators. I. Strictly singular and strictly cosingular operators in C(S)-spaces. Bull. Acad. Pol. Sci. 13, 31–37 (1965)
Plebanek, G.: A construction of a Banach space C(K) with few operators. Preprint, September 2003
Rosenthal, H.: On relatively disjoint families of measures with some applications to Banach space theory. Studia Math. 37, 13–36 (1970)
Rosenthal, H.: The Banach Spaces C(K); Handbook of Geometry of Banach Spaces, Vol 2. Ch. 36 pp. 1547 - 1602 (eds. W.B. Johnson, J. Lindenstrauss); North Holland 2003
Semadeni, Z.: Banach spaces of continuous functions. Państwowe Wydawnictwo Naukowe, 1971
Schachermayer: On some classical measure-theoretic theorems for non-sigma- complete Boolean algebras. Dissertationes Math. (Rozprawy Mat.) 214, (1982)
Talagrand, M.: Un nouveau CE(K) qui possède la propriété de Grothendieck. Israel J. Math. 37, 181–191 (1980)
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While conducting research leading to the results presented in this paper, the author was partially supported by a fellowship Produtividade em Pesquisa from National Research Council of Brazil (Conselho Nacional de Pesquisa, Processo Número 300369/01-8). The final stage of the research was realized at the Fields Institute in Toronto where the author was supported by the State of São Paulo Research Assistance Foundation (Fundação de Amparoá Pesquisa do Estado de São Paulo), Processo Número 02/03677-7 and by the Fields Institute.
Revised version: 29 January 2004
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Koszmider, P. Banach spaces of continuous functions with few operators. Math. Ann. 330, 151–183 (2004). https://doi.org/10.1007/s00208-004-0544-z
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DOI: https://doi.org/10.1007/s00208-004-0544-z