Solutions to several problems in Banach spaces of continuous functions on non-metrizable compact spaces
by
Piotr Koszmider
Universidade de São Paulo

Several fundamental questions concerning the structure of infinite dimensional Banach spaces remained unsolved for many decades. For example, whether every Banach space is decomposable ([Li]), i.e., whether it can be decomposed as A+B into two closed infinite dimensional subspaces, A and B; whether on every Banach space there is an operator other than those of the form cI+C where c is a scalar and C is a compact operator; whether every Banach space has its proper closed subspace which is isomorphic to the entire space ([Ba]), or in particular whether the hyperplanes of a Banach space are isomorphic to the entire space. The first and the third question were answered in the negative only recently (see [Go] and [GM]) and the second remains open.

Our results address similar questions in the context of the classical Banach space C(K) of all continuous functions on a compact Hausdorff space K with the supremum norm. Our results are also negative and depend on set-theoretic topological techniques.

The first construction concerns the question whether there is a Banach space X non-isomorphic to any of its proper subspaces whose particular case for one co-dimensional subspaces, became to be known as the hyperplane problem. For general Banach spaces the problem has been solved only recently by T. Gowers in [Ga] and W. Marciszewski constructed in [Mar] a compact K such that C(K) with the topology of poitwise convergence is not linearly homeomorphic to C(K)xR Various authors (e.g., Semadeni [Se], Lacey [La], Arhangel'skii [Ar]) have posed the hyperplane problem for a particular case of classical Banach spaces of continuous functions on a compact Hausdorff space with the supremum norm.

Our solution addresses the original version of the hyperplane problem that is our C(K) is non-isomorphic to any of its proper subspaces nor to any of its proper quotients. The main ingredient of the proof is showing that for any (linear, bounded) operator T on C(K) its conjugate operator T* has a decomposition gI+S where g is a bounded Borel function on K and S is a weakly compact operator on the space of Radon measures on K.

In the second construction we assume the continuum hypothesis which could be considerably weakened to, for example, Martin's axiom, but we do not know if a special set-theoretic assumption is necessary. The key result concerns the space of bounded linear operators on our C(K), we prove that any such operator is of the form gI+S where g is a continuous function on K and S is weakly compact or equivalently (in the context of C(K) spaces) strictly singular. It is the minimal possible space of operators in C(K) spaces, i.e., weakly compact cannot be replaced with compact.

Since our space K is connected, the result on the space of operators implies the indecomposability of C(K). However one cannot go as far as [GM], i.e., there are no hereditarily indecomposable C(K)'s. The indecomposability of C(K) implies that it is non-isomorphic to any C(K') for K' zero-dimensional, which solves, for example, a problem from [Se] (third problem on page 381, this is also mentioned in [LT]). These problems should be considered in in the context of the classification of separable C(K) i.e., when K is metric, accomplished in the fifties by Miljutin [Mi], Bessaga and Pelczynski [BP]. According to this classification none of the separable C(K) has properties of our examples.

The origin of the method of the construction of K should be traced to [Fe]. However, the final version of our constructions has more similarities with [Ha] and [Ta]. Other new and old results on C(K)'s will be mentioned in the attempt of advertising the Banach space structure of the C(K)'s to mathematicians specializing in compact K's. The preprint(s) can be downloaded from http://www.ime.usp.br/ piotr/CK.html

[Ar] A. V. Arhangel'skii; Problems in C_p-theory; in Open problems in Topology; J. van Mill, G. M. Reed eds. North Holland 1990.

[BP] Cz. Bessaga, A. Pe czy\'nski; Spaces of continuous functions (VI) (On isomorphical classification of spaces C(S)); Studia Math. 19, 1960, pp. 53-62.

[Fe] V. V. Fedorchuk; On the cardinality of hereditarily separable compact Hausdorff spaces; Soviet Math. Dokl. 16, 1975 pp. 651-655.

[Go] W. T. Gowers; A solution to Banach's hyperplane problem; Bull. London Math. Soc. 26 (1994), pp. 523-530.

[GM] W. T. Gowers, B. Maurey; The unconditional basic sequence problem; Journal A. M. S. 6 (1993), pp. 851-874.

[Ha] R. Haydon; A Non-Reflexive Grothendieck Space That Does Not Contain l_\infty; Israel J. Math. Vol. 40, No. 1, 1981, pp. 65-73.

[La] H. E. Lacey; Isometric Theory of Classical Banach Spaces; Springer-Verlag 1974.

[Li] J. Lindenstrauss; Decomposition of Banach spaces; Proceedings of an International Symposium on Operator Theory (Indiana Univ., Bloomington, Ind., 1970). Indiana Univ. Math. J. 20 (1971), no. 10, pp. 917-919.

[LT] J. Lindenstrauss, L. Tzafriri; Classical Banach spaces; Springer Lecture Notes in Math. Vol 338, 1973.

[Mar] W. Marciszewski; A function space C\sb p(X) not linearly homeomorphic to C\sb p(X)×R; Fund. Math. 153, 1997, no. 2, pp. 125-140.

[Mi] A. A. Miljutin; On spaces of continuous functions; Dissertation, Moscow State University, 1952.

[Se] Z. Semadeni; Banach spaces of continuous functions; Pa\'nstwowe Wydawnictwo Naukowe, 1971.

[Ta] M. Talagrand; Un nouveau C(K) qui possède la propriété de Grothendieck; Israel J. Math. 37 (1980), no. 1-2, pp. 181-191.

Date received: February 25, 2003