Banach spaces of continuous functions with few operators
In this paper, the hyperplane problem for \(C(K)\)-spaces is solved in that the author constructs two separable compact spaces \(K_1\) and \(K_2\) such that \(C(K_j)\) is not isomorphic to any proper subspace or quotient of itself, in particular it is not isomorphic to any of its hyperplanes. The first (nonclassical) Banach spaces with these properties were constructed by \textit{W. T. Gowers} and \textit{B. Maurey} [J. Am. Math. Soc. 6, No. 4, 851--874 (1993; Zbl 0827.46008)] and \textit{W. T. Gowers} [Bull. Lond. Math. Soc. 26, No. 6, 523--530 (1994; Zbl 0838.46011)]. The second space \(K_2\), which is considerably harder to construct than the first one, has the additional virtue of being connected; indeed, the complement of each finite subset is connected. This is shown to imply that \(C(K_2)\) is indecomposable, i.e., every complemented subspace of \(C(K_2)\) is either finite-dimensional or finite-codimensional, which solves another long-standing problem. As a consequence, \(C(K_2)\) is not isomorphic to any \(C(L)\) with \(L\) zero-dimensional, and it is the first example of this kind. It is also a Grothendieck space of type \(C(K)\) over a connected~\(K\). The space \(K_1\) arises as the Stone space of a certain Boolean algebra constructed by transfinite induction, and the space \(K_2\) is a subspace of a product \([0,1]^I\). The key feature of the \(C(K)\)-spaces presented in this paper is that all operators \(T: C(K)\to C(K)\) are what the author calls weak multipliers, meaning that \(T^*(\mu)= g\cdot \mu + S(\mu)\) for some bounded Borel function \(g\) and some weakly compact operator \(S:M(K)\to M(K)\). In the final section, it is shown that under the assumption of the continuum hypothesis one can even get examples as above where each operator \(T: C(K)\to C(K)\) itself is a weakly compact perturbation of a multiplication operator. Recently, \textit{G.~Plebanek} [Topology Appl. 143, No. 1--3, 217--239 (2004; Zbl 1064.46011), reviewed below] has obtained the same conclusion without assuming CH; his examples are, however, not separable.
scientific article; zbMATH DE number 2105001
scientific article