Abstract
We say that a Banach space C(K) of all real-valued continuous functions on a compact Hausdorff space K with the supremum norm has few operators if for every linear bounded operator T on C(K) we have T=gI+S or T*=g*I+S where g is continuous on K, g* is Borel on K and S are weakly compact on C(K) or C*(K) respectively. C(K) spaces with few operators share some properties with the spaces of Gowers and Maurey, but their norm is as simple as possible. For example, some of them are indecomposable Banach spaces and are not isomorphic to their hyperplanes. Banach spaces of continuous functions with few operators provided solutions to several long standing open problems in the theory of Banach spaces. This class of spaces is being gradually illuminated and applied further in the recent work of P. Borodulin-Nadzieja, R. Fajardo, V. Ferenczi, E. Medina Galego, M. Martín, J. Merí, G. Plebanek, I. Schlackow and the author. We describe basic properties, applications and relevant open problems concerning this class of Banach spaces.
Resumen
Se dice que un espacio de Banach C(K), formado por las funciones continuas definidas en un espacio compacto de Haussdorf K y con valores reales, provisto con la norma supremo, tiene pocos operadores si para cada operador T en C(K) se tiene que T=gI+S o bien T*=g*I+S, donde la función g: K→K es continua, g*: K→K es una función de Borel y S es débilmente compacto en C(K) o C*(K), respectivamente. Los espacios C(K) con pocos operadores comparten algunas de sus propiedades con los espacios de Gowers y Maurey, pero su norma es la más sencilla posible. Por ejemplo, algunos de ellos son espacios de Banach que no se pueden descomponer y no son isomorfos a sus hiperplanos. Los espacios de Banach de funciones continuas con pocos operadores han proporcionado soluciones a varios problemas de espacios de Banach que han estado abiertos mucho tiempo. Esta clase de espacios está siendo paulatinamente más comprendida y aplicada en artículos recientes de P. Borodulin-Nadzieja, R. Fajardo, V. Ferenczi, E. Medina Galego, M. Martín, J. Merí, G. Plebanek, I. Schlackow y del autor. Aquí describimos propiedades básicas, aplicaciones y relevantes problemas abiertos relativos a esta clase de espacios de Banach.
Similar content being viewed by others
References
Albiac, F. and Kalton, N., (2006). Topics in Banach Space Theory, Springer.
Argyros, S. and Arvanitakis, D., (2004). Regular averaging and regular extension operators in weakly compact subsets of Hilbert spaces, Serdica Math. J., 30, 527–548.
Argyros, S. and Haydon, R., A hereditarily indecomposable L ∞-space that solves the scalar-plus-compact problem, preprint.
Argyros, S.; Lopez-Abad, J. and Todorcevic, S., (2003). A class of Banach spaces with no unconditional basic sequences. Note aux C. R. A. S., Paris, 337, 1.
Bessaga, Cz. and Pełczyński, A., (1960). Spaces of continuous functions (VI) (On isomorphical classification of spaces C(S)), Studia Math., 19, 53–62.
Borodulin-Nadzieja, P., (2007). Measures on minimally generated Boolean algebras, Topology Appl., 154, 3107–3124. DOI: 10.1016/j.topol.2007.03.014
Cook, H., (1966). Continua which admit only the identity mapping onto non-degenerate subcontinua, Fundamenta Math., 60, 241–249.
Casazza, P., (1989). The Schroeder-Bernstein property for Banach spaces, Contemp. Math., 15, 61–78.
Diestel, J. and Uhl, J. J. Jr., (1977). Vector measures, Mathematical Surveys, 15, American Mathematical Society, Providence, R.I.
Dunford,N. and Schwartz, J., (1967). Linear Operators; Part I, General Theory. Interscience Publishers, INC., New York, Fourth printing.
Engelking, R., (1989). General Topology. Helderman Verlag, Berlin.
Fajardo, R., (2007). Consistent constructions of Banach spaces C(K) with few operators. Phd. Thesis, Universidad de São Paulo. (In Portugues).
Fajardo, R., (2009). An indecomposable Banach space of continuous functions which has small density, Fund. Math., 202, 1, 43–63. DOI: 10.4064/fm202-1-2
Fedorchuk, V. V., (1975). On the cardinality of hereditarily separable compact Hausdorff spaces, Soviet Math. Dokl., 16, 651–655.
Ferenczi, V., (1999). Quotient hereditarily indecomposable Banach spaces, Canad. J. Math., 51, 566–584.
Medina Galego, E. and Ferenczi, V., (2007). Even infinite dimensional Banach spaces, J. Funct. Anal., 253(2), 534–549.
Gasparis, I., (2002). A family of continuum totally incomparable hereditarily indecomposable Banach spaces, Studia Math., 151, 277–298.
Gowers, T., (1994). A solution to Banach’s hyperplane problem, Bull. London Math. Soc., 26, 523–530. DOI: 10.1112/blms/26.6.523
Gowers, T., (1996). A Solution to the Schroeder-Bernstein Problem for Banach Spaces, Bull. Lond. Math. Soc., 28(3), 297–304; DOI: 10.1112/blms/28.3.297
Gowers, T. and Maurey, B., (1993). The unconditional basic sequence problem, J. Amer. Math. Soc., 6, 851–874. DOI: 10.2307/2152743
Gowers, T. and Maurey, B., (1997). Banach spaces with small spaces of operators, Math. Ann., 307, 4, 543–568. DOI: 10.1007/s002080050050
Grothendieck, A., (1953). Sur les applications linéaires faiblement compactes d’espaces du type C(K), Canad. J. Math., 5, 129–173.
Haydon, R., (1981). A non-reflexive Grothendieck space that does not contain l ∞, Israel J. Math., 40, 1, 65–73. DOI: 10.1007/BF02761818
Haydon, R.; Odell, E. and Levy, M., (1987). On sequences without weak* convergent convex block subsequences, Proc. Amer. Math. Soc., 100, 1, 94–98. DOI: 10.2307/2046125
Jech, T., (2003). Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin. The third millennium edition, revised and expanded.
Johnson, W. and Lindenstrauss, J., (1974). Some remarks on weakly compactly generated Banach spaces, Israel J. Math., 17, 219–230. DOI: 10.1007/BF02882239 And Israel J. Math., 32 (1979), 4, 382-383. DOI: 10.1007/BF02760467
Koszmider, P., (2004). Banach spaces of continuous functions with few operators, Math. Ann., 330, 1, 151–183. DOI: 10.1007/s00208-004-0544-z
Koszmider, P., (2005). On decompositions of Banach spaces of continuous functions on Mrówka’s spaces, Proc. Amer. Math. Soc., 133, 7, 2137–2146. DOI: 10.1090/S0002-9939-05-07799-3
Koszmider, P., (2005). A space C(K) where all nontrivial complemented subspaces have big densities, Studia Math., 168, 2, 109–127. DOI: 10.4064/sm168-2-2
Koszmider, P., (2009). A C(K) Banach space which does not have the Schroeder-Bernstein property, Preprint.
Koszmider, P.; Martín, M. and Merí, J., (2009). Extremely non-complex C(K) spaces, J. Math. Anal. Appl., 350, 601–615. DOI: 10.1016/j.jmaa.2008.04.021
Koszmider, P.; Martín, M. and Merí, J., (2009). Isometries on extremely non-complex Banach spaces, Accepted to Journal of the Insitute of Mathematics of Jussieu. http://www.institut.math.jussieu.fr/JIMJ/
Kunen, K., (1980). Set Theory. An Introduction to Independence Proofs, North-Holland.
Lacey, E. and Morris, P., (1969). On spaces of the type A(K) and their duals, Proc. Amer. Math. Soc., 23, 1, 151–157. DOI: 10.2307/2037508
Lindenstrauss, J., (1971). Decomposition of Banach spaces, Proceedings of an International Symposium on Operator Theory (Indiana Univ., Bloomington, Ind., 1970). Indiana Univ. Math. J., 20, 10, 917–919.
Maurey, B., (2003). Banach spaces with few operators, in Handbook of Geometry of Banach Spaces, 2. Ch. 29, 1247–1297 (eds. W. B. Johnson and J. Lindenstrauss); North Holland . DOI: 10.1016/S1874-5849(03)80036-0
Pełczyński, A, (1965). On strictly singular and strictly cosingular operators. I. Strictly singular and strictly cosingular operators in C(S)-spaces, Bull. Acad. Pol. Sci., 13, 1, 31–37.
Plebanek, G., (2004). A construction of a Banach space C(K) with few operators, Topology Appl., 143, 217–239. DOI: 10.1016/j.topol.2004.03.001
Plichko, A. and Yost, D., (2000). Complemented and Uncomplemented subspaces of Banach spaces, Extracta Math., 15, 2, 335–371.
Schachermeyer, W., (1982). On some classical measure-theoretic theorems for non-sigma-complete Boolean algebras, Dissertationes Math., CCXIV. PWN (PaństwoweWydawnictwo Naukowe) — Polish Scientific Publisher. Warsaw.
Schlackow, I., (2008). Centripetal operators and Koszmider spaces, Topology Appl., 155, 11, 1227–1236 DOI: 10.1016/j.topol.2008.03.004
Schlackow, I., (2009). Classes of C(K) spaces with few operators D. Phil. Thesis. Univerity of Oxford, available at http://people.maths.ox.ac.uk/schlack1/
Semadeni, Z., (1971). Banach spaces of continuous functions. PWN (Państwowe Wydawnictwo Naukowe) — Polish Scientific Publisher. Warsaw.
Talagrand, M., (1980). Un nouveau C(K) qui possède la propriété de Grothendieck, Israel J. Math., 37, 1–2, 181–191.
Author information
Authors and Affiliations
Corresponding author
Additional information
Submitted by José Orihuela
Rights and permissions
About this article
Cite this article
Koszmider, P. A survey on Banach spaces C(K) with few operators. RACSAM 104, 309–326 (2010). https://doi.org/10.5052/RACSAM.2010.19
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.5052/RACSAM.2010.19