Banach spaces of continuous functions with few operators
type: Item
instance of: scholarly article
title: Banach spaces of continuous functions with few operators
zbMATH Open document ID: 1064.46009
author: Piotr Koszmider
published in: Mathematische Annalen
publication date: 2004-10-01T00:00:00Z
review text: 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 th…
reviewed by: Dirk Werner
Mathematics Subject Classification ID: 46B03
Mathematics Subject Classification ID: 46B25
Mathematics Subject Classification ID: 46B26
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 03E50
Mathematics Subject Classification ID: 06E05
Mathematics Subject Classification ID: 06E15
zbMATH DE Number: 2105001
zbMATH Keywords: indecomposable Banach space
zbMATH Keywords: \(C(K)\)-space
zbMATH Keywords: hereditarily indecomposable space
zbMATH Keywords: hyperplane conjecture
zbMATH Keywords: weak multiplier
zbMATH Keywords: Boolean algebra
zbMATH Keywords: zero-dimensional space
zbMATH Keywords: continuum hypothesis
MaRDI profile type: Publication
full work available at URL: s00208-004-0544-z
OpenAlex ID: W2097649873
DOI: 10.1007/S00208-004-0544-Z
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description: scientific article; zbMATH DE number 2105001