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Incomparable, non-isomorphic and minimal Banach spaces. (English) Zbl 1078.46006

The well-known Gowers, Komorowski and Tomczak-Jaegermann theorem asserts that any Banach space, isomorphic to all its infinite-dimensional subspaces, must necessarily be isomorphic to \(l_2\). It generates many questions about the isomorphic structure of infinite-dimensional subspaces of given Banach space. In particular, W. T. Gowers [Ann. Math. 156, 797–833 (2002; Zbl 1030.46005)] raised the problem to determine which partial order can be realised as the set of subspaces of a Banach space under the relation of isomorphic embeddability.
In the paper under review it is proved that an infinite-dimensional Banach space contains either a minimal subspace or a continuum of pairwise incomparable subspaces. Recall that a Banach space is minimal if it embeds isomorphically into all its infinite-dimensional subspaces. The second theorem is more or less as follows: Let \(X\) be a Banach space with an unconditional basis \((e_n)\). If its set of subspaces admits an ordering under the relation of isomorphic embeddability by real numbers, then any space spanned by a subsequence of \((e_n)\) is isomorphic to its square and its hyperplanes. The proofs essentially use descriptive set theory.

MSC:

46B03 Isomorphic theory (including renorming) of Banach spaces
03E15 Descriptive set theory

Citations:

Zbl 1030.46005
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