X

INTRODUCTION

result originates in the work of Kasin, but the method was developed in [Sz2] and

[S-T].

We als o giv e i n §8. g a n infinite-dimensiona l versio n o f thi s decomposition ,

obtained recently by Krivine.

In Chapter 8, we turn to Banach lattices and start by a reformulation of G.T. in

this context . I n 8.b , w e introduce ultraproduct s wit h severa l simpl e illustrativ e

applications. Problem 2 in the Resume asked whether a specific property (involv-

ing tensor norms) was always satisfied. This was answered negatively by Gordon-

Lewis [G-Ll] . Thei r pape r showe d tha t thi s propert y (no w calle d th e G.L .

property) provide s a usefu l criterio n t o decid e whether o r no t a give n spac e is

isomorphic to a Banach lattice (or more generally to a space with l.u.st.). This is

the subject of 8.c; in 8.d, we show that, for p = £ 2 , the Schatten classes Cp do not

have the G.L. property (cf. [G-Ll, Sc]). Many more spaces without l.u.st. are now

known. Moreover , on e ca n construct , fo r an y n, a n ^-dimensiona l spac e wit h

l.u.st. constan t greate r than S^/n , for som e S 0 independent o f n. This "worst

possible" cas e ca n b e exhibite d i n 8. e ver y quickl y (followin g [F-K-P]) , usin g

Chapter 7. In §8.g, we show (following [L-P]) that an atomic Banach lattice which

satisfies G.T. must be isomorphic to lx(T) fo r some set T.

In Chapte r 9 , we present th e C*-algebrai c versio n of G.T. , a s conjectured b y

Grothendieck. Her e we mainly follow [Pi7 ] and Haagerup' s work [HI]. This was

problem 4 in th e Resume . In §7.b , we discuss (without proofs ) severa l applica -

tions of these results to the theory of derivations and representations of C "'-alge-

bras (cf. [Bu, CI, C2, H2, H3]).

Finally, in Chapter 10, we construct (following [PilO]) several Banach spaces X

such tha t X ® X = X ® X. Thi s give s a negative solutio n t o th e sixt h an d las t

problem i n th e Resume . Grothendieck conjecture d ther e that thi s could happe n

only in the finite-dimensional case. The reader who has reached this point will be

rewarded t o find tha t al l the results used in the construction have been included

(with complet e proofs ) i n th e preceding chapter s (mainl y i n Chapter s 4 , 7 and

6.c).

Each chapter is followed by a notes and references section where the reader will

find th e credit s fo r th e correspondin g results , a s well a s som e additiona l com -

ments. I n general , w e give reference s i n th e tex t itsel f onl y fo r th e statement s

which we quote without proof.

ACKNOWLEDGMENTS.

Thi s is an expanded versio n o f lecture s delivered a t th e

C.B.M.S. Conference hel d in June 1984 at the University of Missouri-Columbia .

It give s me great pleasur e t o have thi s occasion t o than k th e organizers o f thi s

meeting and, in particular, Elias Saab. I am also very grateful t o Nigel Kalton for

his help with the manuscript. I am grateful t o P. Wojtaszczyk an d U . Haagerup

for helpfu l editoria l comments . I woul d lik e t o than k als o Kare n Robinson ,

DeAnna Walkenbach, Karen Brewer, Susan Freie, Suzy Cook, and Regina Teson

for their typing of the preprint version.