Banach spaces without minimal subspaces (Q1028338)

In [``A new dichotomy for Banach spaces'', Geom.\ Funct.\ Anal.\ 6, No.\,6, 1083--1093 (1996; Zbl 0868.46007)], \textit{W.\,T.\thinspace Gowers} proved that every separable infinite-dimensional Banach space \(X\) contains either an unconditional basic sequence or an H.I.\ (hereditarily indecomposable) subspace. He also proved [see \textit{W.\,T.\thinspace Gowers}, ``An infinite Ramsey theorem and some Banach space dichotomies'', Ann.\ Math.\ (2) 156, No.\,3, 797--833 (2002; Zbl 1030.46005)] that every such \(X\) contains either a basic sequence \((x_i)\) such that any two disjointly supported block subspaces are not isomorphic or a quasiminimal subspace \(Y\) (for any two infinite-dimensional subspaces, some infinite-dimensional Banach space \(Z\) embeds into both). The authors call these results Gowers' first and second dichotomy theorems, respectively. The paper under review continues the study of the search for necessary subspaces with more dichotomy theorems. Henceforth, all \(X,Y,Z,\dots\) will be separable infinite-dimensional Banach spaces. \(X\) is minimal if for all \(Y\subseteq X\), \(X\) embeds into \(Y\). Their third dichotomy theorem is that if \(X\) contains no minimal subspace, then it contains a tight subspace, \(E\). This means that \(E\) has a basis \((e_i)\) with the property that for any space \(Y\) there exists a sequence \(I_0< I_1<\dots\) of successive nonempty intervals in \(\mathbb N\) so that if \(A\subseteq \mathbb N\) and if \(Y\) embeds into \([(e_i):i\in \bigcup_{i\in A} I_i]\), then \(A\) is co-finite. The proof uses infinite asymptotic games in Banach spaces. It is also shown that a tight space contains no minimal subspaces. A basis \((e_i)\) is tight with constants if in the definition of tightness, given \(Y\), one may choose the successive intervals \(I_0 < I_1 <\dots\) so that \(Y\) does not \(K\)-embed into \([e_i:i\notin I_K]\). Tsirelson's space \(T\) is shown to be an example of such a space. A space \(X\) is locally minimal if for some \(K\), \(X\) is \(K\)-crudely finitely representable in every subspace. Their fifth dichotomy theorem is that every basis \((e_n)\) admits a block basis \((x_n)\) that is either tight with constants or \([(x_n)]\) is locally minimal. A tight basis \((e_n)\) is tight by range if, whenever \((y_n)\) is a block subspace of \((e_n)\), the sequence \((I_n)\) of successive intervals that witness the tightness of \(Y\) can be chosen by \(I_n = \text{range }y_n\) (w.r.t.\ \((e_i)\)). A basis \((x_n)\) for \(X\) is subsequently minimal if every subspace of \(X\) contains an isomorph of some \([(x_{n_i})_{i=1}^\infty]\). \(T\) is such a space. These last two properties are shown to be exclusive. The fourth dichotomy theorem is that every \(X\) contains a basis \((e_n)\) that is either tight by range or subsequentially minimal. The paper contains many more results, such as dichotomy theorems for the cases when \(X\) contains an asymptotic \(\ell_p\) subspace for some \(p\) or not, and for the cases when \(X\) contains an isomorph of some \(\ell_p\) or \(c_0\) or not. Finally, the paper contains a descriptive set theory section concerning the partial order induced on the bi-embeddability classes of subspaces of a space \(X\) by isomorphic embeddability, and a concluding long table of necessary basic sequences that must be found inside every \(X\).