Factorization of linear operators and geometry of Banach spaces. (English) Zbl 0588.46010

Pisier, Gilles

Regional Conference Series in Mathematics 60. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-0710-2). x, 154 p. (1986).

This paper is essentially a report on developments in the theory of Banach space structure which have occurred since the appearance in 1968 of the paper of J. Lindenstrauss and A. Pelczynski [Stud. Math. 29, 275-326 (1968; Zbl 0183.405)] which first gave explicit form and application to the ideas inherent in A. Grothendieck’s Resumé de la theorie metrique des produits tensoriels topologiques [Bol. Soc. Mat. Sao-Palo 8, 1-79 (1956; Zbl 0074.323)]. These developments revolve around properties of factorization of operators through certain Banach spaces, in particular Hilbert space, and the implications that such factorizations (or their impossibility) have for Banach space structure.
Though the richness and diversity of the results contained in the paper preclude their detailed description, a listing of the Table of Contents gives a fair summary of the scope of the material included: Chapter 0. Preliminary Results and Background; Chapter 1. Absolutely Summing Operators and Basic Applications; Chapter 2. Factorization trough a Hilbert Space; Chapter 3. Type and Cotype. Kwapien’s Theorem: Chapter 4. The ”Abstract” Version of Grothendieck’s Theorem; Chapter 5. Grothendieck’s Theorem; Chapter 6. Banach Spaces Satisfying Grothendieck’s Theorem; Chapter 7. Applications of the Volume Ratio Method; Chapter 8. Banach Lattices; Chapter 9. \(C^*\)-Algebras; Chapter 10. Counterexamples to Grothendieck’s Conjecture.
The paper is very carefully and understandably written with little presupposed that is not at least mentioned in passing or referenced in an extensive bibliography.

Reviewer: J.R.Holub

MSC:

46B20	Geometry and structure of normed linear spaces
47A68	Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators
47B10	Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
46M05	Tensor products in functional analysis
47C15	Linear operators in \(C^*\)- or von Neumann algebras
46L05	General theory of \(C^*\)-algebras

Keywords:

Type; Cotype; Grothendieck’s Theorem; Volume Ratio Method; Banach Lattices; \(C^*\)-Algebras; Counterexamples to Grothendieck’s Conjecture

Citations:

Zbl 0183.405; Zbl 0074.323