Spaces of measurable vector-functions not containing the space \(\ell^ 1\) (Q793287)

Let \((T,\Sigma\),\(\mu)\) be a \(\sigma\)-finite measure space and E be an ideal Banach space of measurable functions on T. For a Banach space X denote by E(X) the space of all measurable functions z:\(T\to X\) such that \(\| z(\cdot)\|_ X\in E\), with the norm \(\| z\| =\| \| z(\cdot)\|_ X\|_ E.\) The authors prove that for \(E\neq \{0\}\) and \(X\neq \{0\}\) the Banach space E(X) does not contain subspaces isomorphic to \(\ell^ 1\) if and only if E and X do not contain subspaces isomorphic to \(\ell^ 1\). This gives a positive answer to a problem of A. V. Bukhvalov [Usp. Mat. Nauk 35, No.1(211), 235 (1980)]. In the case \(E=L^ p\) the theorem was proved by \textit{G. Pisier} [C. R. Acad Sci., Paris, Ser. A 286, 747-749 (1978; Zbl 0373.46033)].