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Symmetric function spaces on atomless probability spaces. (English) Zbl 0715.46011

The work under review is concerned with various more or less separate topics on symmetric function spaces. Here is a selection of what the authors have proved.
Within preliminary sections (Sections 1-3) they provide a new proof of the Maharam’s theorem. In what follows, E stands for a symmetric function space over an atomless probability space.
Section 4: Bases in nonseparable symmetric spaces. The authors prove that the Walsh system in E with absolutely continuous norm forms an Enflo- Rosenthal-Cesàro basis (Theorem 1). Further properties of the Walsh system in E are provided in Theorem 2; they depend upon the Boyd indexes.
Section 5: Uncountable unconditional basis sequences and unconditional decompositions. Main result: For E with absolutely continuous norm, every subspace of E has a monotone unconditional projection decomposition provided that the upper Boyd index is finite and the lower one is greater than 1. This generalizes a result of W. B. Johnson and E. Odell [Compositio Math. 28, 37-49 (1974; Zbl 0282.46020)], where \(E=L_ p(\mu)\) with \(1<p<\infty.\)
Section 6: Characters of compact Abelian groups. Theorem 3: The character group \(G'\) of an arbitrary compact Abelian group G can be decomposed into countably many subsets each of which is equivalent in every \(L_ p(G)\) \((1\leq p<\infty)\) to an orthonormal basis of a Hilbert space.
Section 7: Almost periodic functions. The notion of a symmetric space of almost priodic functions is introduced by extending Basicovitch spaces and some of their generalizations, and the previous results are applied to these spaces.
Section 8: Narrow operators. Narrow operator are characterized (another terminology: norm-sign-preserving operators) and a class of symmetric spaces on which there is no non-zero narrow operator is exhibited.
Section 9: Narrow operators in \(L_ p(\mu)\), \(1\leq p<\infty\). Theorem 7: Let T be a narrow operator from \(L_ p(\mu)\) into itself \((1<p<\infty\), \(p\neq 2)\). Then for each \(\epsilon >0\) there is \(\delta_ p(\epsilon)>0\) with \(\| I+T\| \geq 1+\delta_ p(\| T\|)\) where I is the identity operator of \(L_ p(\mu)\). This was known previously for T a compact self-operator of \(L_ p(0,1).\)
Section 10: Rich subspaces.
Section 11: Isomorphic classification of \(L_ p(\mu)\)-spaces. Theorem 1: With \(0<p<1\), \(L_ p(\mu)\) and \(L_ p(\nu)\) are isomorphic iff the Maharam set of measures \(\mu\) and \(\nu\) coincide. Theorem 2: If \(1\leq p<\infty\), \(p\neq 2\), and the Maharam sets of \(\mu\) and \(\nu\) are distinct, then the Banach-Mazur distance between \(L_ p(\mu)\) and \(L_ p(\nu)\) is \(\geq 2\) if \(p=1\), and is \(\geq 1+\delta_ p(1)\) otherwise.
Reviewer: T.Kubiak

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces

Citations:

Zbl 0282.46020