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Primariness of some vector-valued function spaces. (English) Zbl 0631.46039

A linear topological space X is called primary if \(X=Y\oplus Z\) implies \(Y\approx X\) or \(Z\approx X\). For a separable infinite dimensional rearrangement invariant Banach space X on \(I=[0,1]\) such that the Boyd indices \(p_ X\), \(q_ X\) satisfy \(1<p_ X\leq q_ X<\infty\) and a Banach space Y with a symmetric basis put \(X(Y)=\{f: I\to Y\); \(g(t):=\| f(t)\|_ Y\in X\}\) normed by \(\| f\|_{X(Y)}=\| \| f(t)\|_ Y\|_ X.\)
The author raises the following problem: Is the Banach space X(Y) primary?
In this paper he gives a partial answer to this question proving that X(Y) is primary in two cases: a) if X,Y are as above and further X does not contain uniformly \(\ell^ 1(n)\) and Y is s-convex and r-concave for \(1<s\leq r<\infty\); b) \(X=L_{p,q}\), \(1<p<\infty\), \(0<q<\infty\) and Y is as in a) and satisfies a suplementary condition called \({\mathcal P}_ X\). The results of this paper extend some results of D. Alspach, P. Enflo and E. Odell, Stud. Math. 60, 79-90 (1977).
Reviewer: C.Mustăţa

MSC:

46E40 Spaces of vector- and operator-valued functions
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B25 Classical Banach spaces in the general theory
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