Banach spaces and descriptive set theory: selected topics.

*(English)*Zbl 1215.46002
Lecture Notes in Mathematics 1993. Berlin: Springer (ISBN 978-3-642-12152-4/pbk; 978-3-642-12153-1/ebook). xi, 161 p. (2010).

The author uses descriptive set theory to prove results on the structure of Banach spaces. More precisely, he is concerned with universality problems, and treats questions of the following type:

Let \({\mathcal C}\) be a class of separable Banach spaces such that every space \(X\) in the class \({\mathcal C}\) has a certain property, say property \((P)\). When can we find a separable Banach space \(Y\) which has property \((P)\) and contains an isomorphic copy of every member of \({\mathcal C}\)?

It is shown that this is possible, for various properties \((P)\), such as: “being reflexive”, “having a separable dual”, “not containing an isomorphic copy of \(c_0\)” (or, more generally, “not containing a minimal space not containing \(\ell_1\)”), or “being non-universal”, if one assumes that the class \({\mathcal C}\) is analytic (for the Effros-Borel topology on the set of all subspaces of \(C ([0,1])\)). This is proved in Chapter 7, with the previous chapters providing the material to reach it.

The idea of the construction is the following. Starting with the data \({\mathcal X}\) of a Banach space \(X\), a tree \(T\) (more precisely, a pruned tree) on a denumerable set, and a kind of “basis” \((x_t)_{t \in T}\) of \(X\) indexed by \(T\), one can construct an \(\ell_2\)-Schauder tree basis \(T^{\mathcal X}_2\) as the completion of \(c_{00} (T)\) equipped with a norm whose definition is analoguous to that of the James tree space. Such a construction was used by J. Bourgain in 1980 in a particular case to show that every separable Banach space which is universal for all reflexive Banach spaces is actually universal for all separable Banach spaces [“On separable Banach spaces, universal for all separable reflexive spaces,” Proc. Am. Math. Soc. 79, 241–246 (1980; Zbl 0438.46005)]. A general construction was afterwards given by B. Bossard in his thesis in 1994, but only published in 2002 [“A coding of separable Banach spaces. Analytic and coanalytic families of Banach spaces,” Fundam. Math. 172, No. 2, 117–152 (2002; Zbl 1029.46009)], but only for the tree of all finite sequences of integers. Here, one must work with more general trees, and some complications occur: though, in “good” cases (when the tree is well-founded) one can generate \(\ell_2\)-hereditary Banach spaces, it happens that “often” one has “many” subspaces of \(T^{\mathcal X}_2\) containing \(c_0\). In order to “control” all the subspaces, the space \(T^{\mathcal X}_2\) must be replaced by a more complicated notion: the notion of amalgamated spaces. This is done in Chapter 4. This notion was introduced by S. Argyros and P. Dodos [“Genericity and amalgamation of classes of Banach spaces,” Adv. Math. 209, No. 2, 666–748 (2007; Zbl 1109.03047)]. They are \(p\)-interpolation spaces, in the sense of Davis-Figiel-Johnson-Lindenstrauss, of \(T^{\mathcal X}_2\). The universal spaces obtained in Chapter 7 are amalgamated spaces obtained from the universal space \(U\) of A. Pełczýnski [Stud. Math. 32, 247–268 (1969; Zbl 0185.37401)]. But in order to construct some of these spaces, it is necessary to revisit some difficult previous constructions, in order to “code” them in a suitable manner. The first one is Zippin’s construction [M. Zippin, Trans. Am. Math. Soc. 310, No. 1, 371–379 (1988; Zbl 0706.46015)], saying that every Banach space with a separable dual embeds in a Banach space with a shrinking basis. This had actually been done by B. Bossard in his thesis, but more consequences are needed. This is the object of Chapter 5. The second construction is that of J. Bourgain and G. Pisier [Bol. Soc. Bras. Mat. 14, No. 2, 109–123 (1983; Zbl 0586.46011)] for \({\mathcal L}_\infty\)-spaces. This is the object of Chapter 6. In Chapter 7, all this material is used to prove the announced results; for that, the concept of strongly bounded class of separable Banach spaces is introduced as a central concept.

The contents of the book are as follows.

Chapter 1: Basic concepts (Polish spaces and standard Borel spaces, trees, universal spaces); Chapter 2: The standard Borel space of all separable Banach spaces; Chapter 3: The \(\ell_2\) Baire sum; Chapter 4: Amalgamated spaces; Chapter 5: Zippin’s embedding theorem; Chapter 6: The Bourgain-Pisier construction; Chapter 7: Strongly bounded classes of Banach spaces; Appendix A: Rank theory; Appendix B: Banach space theory; Appendix C: The Kuratowski-Tarski algorithm; Appendix D: Open problems.

In conclusion, I think that, besides the results, this book may be useful for people interested in Banach space theory or/and descriptive set theory. It is very well written and contains a lot of results and techniques from these two theories, and thus may serve as a reference book.

Let \({\mathcal C}\) be a class of separable Banach spaces such that every space \(X\) in the class \({\mathcal C}\) has a certain property, say property \((P)\). When can we find a separable Banach space \(Y\) which has property \((P)\) and contains an isomorphic copy of every member of \({\mathcal C}\)?

It is shown that this is possible, for various properties \((P)\), such as: “being reflexive”, “having a separable dual”, “not containing an isomorphic copy of \(c_0\)” (or, more generally, “not containing a minimal space not containing \(\ell_1\)”), or “being non-universal”, if one assumes that the class \({\mathcal C}\) is analytic (for the Effros-Borel topology on the set of all subspaces of \(C ([0,1])\)). This is proved in Chapter 7, with the previous chapters providing the material to reach it.

The idea of the construction is the following. Starting with the data \({\mathcal X}\) of a Banach space \(X\), a tree \(T\) (more precisely, a pruned tree) on a denumerable set, and a kind of “basis” \((x_t)_{t \in T}\) of \(X\) indexed by \(T\), one can construct an \(\ell_2\)-Schauder tree basis \(T^{\mathcal X}_2\) as the completion of \(c_{00} (T)\) equipped with a norm whose definition is analoguous to that of the James tree space. Such a construction was used by J. Bourgain in 1980 in a particular case to show that every separable Banach space which is universal for all reflexive Banach spaces is actually universal for all separable Banach spaces [“On separable Banach spaces, universal for all separable reflexive spaces,” Proc. Am. Math. Soc. 79, 241–246 (1980; Zbl 0438.46005)]. A general construction was afterwards given by B. Bossard in his thesis in 1994, but only published in 2002 [“A coding of separable Banach spaces. Analytic and coanalytic families of Banach spaces,” Fundam. Math. 172, No. 2, 117–152 (2002; Zbl 1029.46009)], but only for the tree of all finite sequences of integers. Here, one must work with more general trees, and some complications occur: though, in “good” cases (when the tree is well-founded) one can generate \(\ell_2\)-hereditary Banach spaces, it happens that “often” one has “many” subspaces of \(T^{\mathcal X}_2\) containing \(c_0\). In order to “control” all the subspaces, the space \(T^{\mathcal X}_2\) must be replaced by a more complicated notion: the notion of amalgamated spaces. This is done in Chapter 4. This notion was introduced by S. Argyros and P. Dodos [“Genericity and amalgamation of classes of Banach spaces,” Adv. Math. 209, No. 2, 666–748 (2007; Zbl 1109.03047)]. They are \(p\)-interpolation spaces, in the sense of Davis-Figiel-Johnson-Lindenstrauss, of \(T^{\mathcal X}_2\). The universal spaces obtained in Chapter 7 are amalgamated spaces obtained from the universal space \(U\) of A. Pełczýnski [Stud. Math. 32, 247–268 (1969; Zbl 0185.37401)]. But in order to construct some of these spaces, it is necessary to revisit some difficult previous constructions, in order to “code” them in a suitable manner. The first one is Zippin’s construction [M. Zippin, Trans. Am. Math. Soc. 310, No. 1, 371–379 (1988; Zbl 0706.46015)], saying that every Banach space with a separable dual embeds in a Banach space with a shrinking basis. This had actually been done by B. Bossard in his thesis, but more consequences are needed. This is the object of Chapter 5. The second construction is that of J. Bourgain and G. Pisier [Bol. Soc. Bras. Mat. 14, No. 2, 109–123 (1983; Zbl 0586.46011)] for \({\mathcal L}_\infty\)-spaces. This is the object of Chapter 6. In Chapter 7, all this material is used to prove the announced results; for that, the concept of strongly bounded class of separable Banach spaces is introduced as a central concept.

The contents of the book are as follows.

Chapter 1: Basic concepts (Polish spaces and standard Borel spaces, trees, universal spaces); Chapter 2: The standard Borel space of all separable Banach spaces; Chapter 3: The \(\ell_2\) Baire sum; Chapter 4: Amalgamated spaces; Chapter 5: Zippin’s embedding theorem; Chapter 6: The Bourgain-Pisier construction; Chapter 7: Strongly bounded classes of Banach spaces; Appendix A: Rank theory; Appendix B: Banach space theory; Appendix C: The Kuratowski-Tarski algorithm; Appendix D: Open problems.

In conclusion, I think that, besides the results, this book may be useful for people interested in Banach space theory or/and descriptive set theory. It is very well written and contains a lot of results and techniques from these two theories, and thus may serve as a reference book.

Reviewer: Daniel Li (Lens)

##### MSC:

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

46B03 | Isomorphic theory (including renorming) of Banach spaces |

46B07 | Local theory of Banach spaces |

46B15 | Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces |

05D10 | Ramsey theory |

03E15 | Descriptive set theory |

54H05 | Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) |