Beiträge zur Strukturtheorie der Grothendieck-Räume. (Contributions to the structure theory of Grothendieck spaces). (German) Zbl 0597.46021
Sitzungsber., Heidelberger Akad. Wiss., Math.-Naturwiss. Kl. 4, 78 S. (1985).
A Banach space E is called a Grothendieck space whenever every \(\sigma\) (E’,E) - convergent sequence in E’ is \(\sigma\) (E’,E”) - convergent. The author presents in this booklet a nice survey of known results on Grothendieck spaces and adds some extensions and new results. Without going into details, we list some of the major topics.
(I) In section 8 some sufficient conditions on the ordering in a Banach lattice E are given in order that \(\sigma\) (E’,E) - convergence is identical to \(\sigma (E',I_ E)\)- convergence \((I_ E\) denotes the order ideal generated by E in E”). In section 10 the same is done for the Grothendieck property.
(II) It is well-known that quotients of Grothendieck spaces cannot be isomorphic to \((c_ 0)\). A Grothendieck space cannot contain therefore a complemented subspace, isomorphic to \((c_ 0)\). In sections 2, 3 and 7 it is investigated in how far the converse of these and similar results holds for Banach lattices. It turns out that a lattice theoretical analogue of property (V) of Pełczyński plays an important rôle in this respect (section 6). The considerations of sections 6 and 7 lead to generalizations of results due to Niculescu, Figiel, Ghoussoub, Johnson and Kühn.
(III) Section 11 deals mainly with the Grothendieck property of \({\mathcal F}\)-products and \(\ell_{\infty}\)-direct sums of Banach spaces and Banach lattices.
(IV) In section 4, finally, a generalization is given of a characterization of Grothendieck spaces of type C(K) (K compact), due to Diestel and Seifert. The author shows that this result can be extended to a class of Banach spaces which includes all complemented subspaces of Banach lattices and all quotients of Lindenstrauss spaces.
Some marginal notes: in the first place, the present proof of theorem 5.1, implication (III) \(\to\) (II), seems to be wrong without the extra assumption, though the result is correct. Secondly, it is in many places not clear which results are due to the author and which results are taken from the literature.
(I) In section 8 some sufficient conditions on the ordering in a Banach lattice E are given in order that \(\sigma\) (E’,E) - convergence is identical to \(\sigma (E',I_ E)\)- convergence \((I_ E\) denotes the order ideal generated by E in E”). In section 10 the same is done for the Grothendieck property.
(II) It is well-known that quotients of Grothendieck spaces cannot be isomorphic to \((c_ 0)\). A Grothendieck space cannot contain therefore a complemented subspace, isomorphic to \((c_ 0)\). In sections 2, 3 and 7 it is investigated in how far the converse of these and similar results holds for Banach lattices. It turns out that a lattice theoretical analogue of property (V) of Pełczyński plays an important rôle in this respect (section 6). The considerations of sections 6 and 7 lead to generalizations of results due to Niculescu, Figiel, Ghoussoub, Johnson and Kühn.
(III) Section 11 deals mainly with the Grothendieck property of \({\mathcal F}\)-products and \(\ell_{\infty}\)-direct sums of Banach spaces and Banach lattices.
(IV) In section 4, finally, a generalization is given of a characterization of Grothendieck spaces of type C(K) (K compact), due to Diestel and Seifert. The author shows that this result can be extended to a class of Banach spaces which includes all complemented subspaces of Banach lattices and all quotients of Lindenstrauss spaces.
Some marginal notes: in the first place, the present proof of theorem 5.1, implication (III) \(\to\) (II), seems to be wrong without the extra assumption, though the result is correct. Secondly, it is in many places not clear which results are due to the author and which results are taken from the literature.
Reviewer: C.B.Huijsmans
