Math. Ann. 284, 41 53 (1989) mathematisclm Annalen Springer-Verlag 1989 Grothendieck's theorem and factorization of operators in Jordan triples Cho-Ho Chu 1, Bruno Iochum 2, and Guy Loupias 3 Goldsmiths' College, London SE14 6NW, England 2 Universit6 de Provence and Centre de Physique Th~orique CNRS*, F-13288 Marseille Cedex 9, France 3 Laboratoire de Physique Math6matique**, USTL, F-34060 Montpellier, France 1. Introduction Recently Pisier [23, 25] and Haagerup [9] proved a noncommutative version of Grothendieck's theorem [8] for C*-algebras which has many applications. This theorem has now been further extended by Barton and Friedman [2] to a class of complex Banach spaces, the so-called JB*-triples, which includes the C*-algebras but need not have any order structure. In connection with Grothendieck's theorem, Pisier [26] has also studied the factorization of operators from C*- algebras to Banach spaces of finite cotype. This paper arose from an attempt to understand the different proofs of Grothendieck's theorem in different contexts and to extend Pisier's recent work in [26] on the factorization of (q, p)-summing operators to JB*-triples. We will give an alternative approach to Grothendieck's theorem for JB*- triples and prove some factorization theorems for operators from JB*-triples to Banach spaces of finite cotype which generalize Pisier's results [26] for C*- algebras. Some applications will also be discussed. It has been shown in [4] that a JBW*-triple is a complemented subspace of JB W*-algebra. Using this result and the fact that Grothendieck's theorem for C*- algebras can be carried over verbatim to JB*-algebras, one obtains Grothendieck's theorem for JB*-triples. This approach relies more on the general properties of Banach spaces but does not involve detailed computation with the "triple arithmetic" and hence gives a less accurate estimate of Grothendieck's constant than that of Barton and Friedman [2]. We should point out, however, that both approaches for Jordan triples hinge on the ideas of Pisier and Haagerup. In [4] it was proved that a finite cotype quotient of a JB*-triple J is of type 2. Thus, if S:J~Y is a bounded operator onto a cotype 2 Banach space Y, then J/Ker S is both type 2 and cotype 2 which is therefore isomorphic to a Hilbert space by Kwapien's result [18]. In particular, S factors through a Hilbert space. In fact, we will show that the factorization still holds even if S is not onto; but this is a * Laboratoire propre du CNRS ** Unit6 associ~e au CNRS n ~ 040768