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A construction of a Banach space \(C(K)\) with few operators. (English) Zbl 1064.46011

In [Math.Ann.330, No.1, 151–183 (2004; Zbl 1064.46009), reviewed above], P. Koszmider solved (in ZFC) the hyperplane problem for \(C(K)\)-spaces in the negative. His approach was to construct compact spaces \(K\) for which all operators on \(C(K)\) are of a special type called weak multipliers. In fact, assuming the continuum hypothesis he was able to construct (even connected) compact spaces \(K\) for which every operator on \(C(K)\) is the sum of a multiplication operator and a weakly compact operator. In this paper, such a compact space is called a Koszmider space.
The purpose of the present paper is to remove CH from the construction of Koszmider spaces. The author develops a topological property for \(K\) to be a Koszmider space, and he constructs connected Koszmider spaces purely in ZFC. These arise as representation spaces of certain lattices. The examples constructed here are, however, not separable as opposed to Koszmider’s.
The paper is very clearly written; but the reader should be cautioned that the author apparently refers to a preprint version of Koszmider’s work. So weak multipliers are called centripetal operators here, and cross-referencing does not seem to match the published version.

MSC:

46B03 Isomorphic theory (including renorming) of Banach spaces
46B25 Classical Banach spaces in the general theory
46B26 Nonseparable Banach spaces
54C35 Function spaces in general topology
54D30 Compactness
54H10 Topological representations of algebraic systems
03E50 Continuum hypothesis and Martin’s axiom
06E05 Structure theory of Boolean algebras
06E15 Stone spaces (Boolean spaces) and related structures

Citations:

Zbl 1064.46009
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References:

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