Ramsey methods in analysis
The book under review has two parts. The first part ``Saturated and unconditional structures in Banach spaces'' is written by Spiros A. Argyros and presents constructions of mixed Tsirelson and hereditarily indecomposable (HI) spaces. The second part ``High-dimensional Ramsey theory and Banach space geometry'' is written by Stevo Todorcevic and presents connections between Ramsey theory and Banach space geometry. The first part of the book has eight chapters. Chapter~I discusses Tsirelson spaces and mixed Tsirelson spaces. Tsirelson space was the first known example of a Banach space that does not contain any isomorph of \(\ell_p\) (\(1 \leq p < \infty\)) or \(c_0\). The mixed Tsirelson spaces provide a generalization of the classical Tsirelson space and they were introduced and studied by Argyros and Deliyanni. Chapter~II presents the mixed Tsirelson extensions of a ground norm. These were introduced by Argyros and Tolias. Starting from a set \(G\) (which satisfies very mild assumptions and is called a \textit{ground set}) which is a subset of \(c_{00} (\kappa )\), the finitely supported sequences on an ordinal \(\kappa\), one defines a norm \(\| { \cdot \| _G}\) on \(c_{00} (\kappa )\) so that \(G\) is a norming set and denotes by \(Y_G\) its completion. The norm of the mixed Tsirelson extension \(T_{\kappa}[G]\) of \(Y_G\) is defined as an increasing limit of a sequence of norms which are defined inductively having \({\| \cdot \| _G}\) as the starting point. If \(T_{\kappa}[G]\) is a strictly singular extension of \(Y_G\) (i.e., if the identity map from \(T_{\kappa}[G]\) to \(Y_G\) is strictly singular) then \(T_{\kappa}[G]\) is reflexive and unconditionally saturated. Moreover \(T_{\kappa}[G]\) does not contain any \(\ell_p\) (\(1 \leq p < \infty\)). The transfinite basis of \(T_{\kappa}[G]\) is boundedly complete, hence \(T_{\kappa}[G]\) is a dual space. The tree complete extension \({{\mathfrak X}} [ G, \sigma]\) of \(Y_G\) was introduced by Argyros and Tolias. The construction depends on a coding function \(\sigma\) which has its origins in work of Maurey and Rosenthal. In chapter~III it is proved that if \({{\mathfrak X}} [ G , \sigma]\) is a strictly singular extension of \(Y_G\) then it is an HI space and called an HI extension of \(Y_G\). Also the HI property of the predual \({{\mathfrak X}} [ G , \sigma]_*\) of \({{\mathfrak X}} [ G , \sigma]\) is examined. In chapter~IV the space of operators on \({{\mathfrak X}} [ G , \sigma]\) and \({{\mathfrak X}} [ G , \sigma ]_*\) is examined. Chapter~V gives three examples of HI extensions due to Argyros and Tolias. The first example is a quasi-reflexive HI space. The second example is the HI space \({{\mathfrak X}} _p\) (\(1 < p < \infty\)) which has \(\ell_p\) as a quotient. The third example is a non-separable HI space. Chapter~VI presents the Banach space \({{\mathfrak X}} _{\omega_1}\), a non-separable reflexive space having a transfinite basis \((e_\alpha)_{\alpha < \omega_1}\) and containing no unconditional basic sequence. The space \({{\mathfrak X}} _{\omega_1}\) was constructed by Argyros, Lopez-Abad and Todorcevic. Chapter~VII examines the space of diagonal non-strictly singular operators on \({{\mathfrak X}} _{\omega_1}\), and Chapter~VIII the bounded linear operators from a subspace of \({{\mathfrak X}} _{\omega_1}\) to \({{\mathfrak X}} _{\omega_1}\). The second part of the book has four chapters. Chapter~I presents the finite dimensional Ramsey theory. It starts with the classical theorem of Ramsey on integers. As an application, the result of Brunel and Sucheston is presented and the property of finite representability of Banach spaces is then discussed. In particular, the theorem of Krivine is proved. A discussion about ultrapowers of Banach spaces corresponding to non-trivial ultrafilters is also included. Chapter~II presents the Ramsey theory of finite and infinite sequences. The theorem of Kruskal is proved. Nash-Williams's theory of fronts and barriers and uniform fronts and barriers is presented. The theorem of Erdős-Rado and the generalization of this theorem to uniform barriers is included. Applications of these results are presented for extracting unconditional subsequences of weakly null sequences in certain Banach spaces. The theorem of Galvin-Prikry is proved, and as an application Farahat's proof of Rosenthal's \(\ell_1\)-theorem is given. Laver's proof of Fraïssé's conjecture is proved. Also Ellentuck's characterization of subsets of \({{\mathbb N}}^{[\infty]}\) having the Ramsey property is given. Applications on summability in Banach spaces and in topological abelian groups are presented. Chapter~III presents the Ramsey theory of finite and infinite block sequences. The proofs of Hindman's and Milliken's results are included. Finally, Gowers's result on the oscillation of real valued Lipschitz functions on the unit sphere of \(c_0\) is proved. The final chapter, chapter IV, presents the approximate and strategic Ramsey theory of Banach spaces. The famous Gowers dichotomy and Gowers's rough classification of Banach spaces (stating that if \(X\) is a Banach space with a Schauder basis then either \(X\) has a quasi-minimal subspace or it has a block subspace \(Y\) such that every two disjointly supported subspaces of \(Y\) are totally incomparable) are proved. The book is carefully written with clear exposition of the material. It can be studied by graduate students who had a first course in functional analysis and are interested in either functional analysis or Ramsey theory.
hereditarily indecomposable Banach spaces
scientific article; zbMATH DE number 2174318