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.
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