Conference Board of the Mathematical Sciences REGIONAL CONFERENCE SERIES IN MA THEM A TICS

supported by the National Science Foundation

Number 60

FACTORIZATION OF LINEAR OPERATORS AND GEOMETRY OF BANACH SPACES

Gilles Pisier

Published for the Conference Board of the Mathematical Sciences

by the American Mathematical Society

Providence, Rhode Island

http://dx.doi.org/10.1090/cbms/060

Expository Lecture s from th e CBM S Regional Conferenc e

held a t th e Universit y o f Missouri-Columbi a June 25-29 , 198 4

Research supporte d i n par t b y Nationa l Scienc e Foundatio n Gran t DM S 84-01302.

1980 Mathematics Subject Classifications (198 5 Revision). Primar y 46B99 , 47B10,

46M05; Secondary 46B30 , 46J15, 46C99, 46L30.

Library o f Congres s Cataloging-in-Publication Dat a

Pisier, Gilles , 1950 -Factorization o f linea r operator s an d geometr y o f Banac h spaces . (Regional conferenc e serie s in mathematics ; no. 60) "Lectures presente d a t th e NSF-CBM S regional conference , Universit y o f Missouri -

Columbia, June 25-29 , 1984" -

Includes bibliographies . L Linea r operators . 2 . Factorizatio n o f operators . 3 . Banac h spaces . I . Conferenc e

Board o f th e Mathematica l Sciences . II . Title . III . Series . QA1.R33 no . 60 510 s [515.7'246 ] 81-1860 5

[QA329.2] ISBN 0-8218-0710-2 (alk . paper )

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FACTORIZATION OF LINEAR OPERATORS AND GEOMETRY OF BANACH SPACES

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Contents

Introduction vi i

Chapter 0. Preliminary Results and Background 1 a. Genera l notatio n 1 b. An introductio n t o tenso r products . The approximatio n property .

Nuclear operator s 2 c. Loca l reflexivity 6

Chapter 1 . Absolutely Summing Operators and Basic Applications 9 a. Absolutely summin g operators 9 b. Applications t o Banach spaces 1 6 c. An introductio n t o duality theory . Integral operators 1 8

Notes and reference s 1 9

Chapter 2. Factorization through a Hilbert Space 2 1 a. Operators factorin g throug h a Hilbert spac e 2 1 b. A duality theore m 2 5

Notes an d reference s 3 0

Chapter 3. Type and Cotype. Kwapieh's Theorem 3 1 a. Type and cotype . Definitions 3 1 b. Kwapieh's theore m 3 2 c. Supplementar y result s 3 4 d. Type and cotyp e and th e geometry of Banac h spaces 3 8

Notes and reference s 4 0

Chapter 4. The "Abstract" Version of Grothendieck's Theorem 4 1 a. The factorization theore m 4 1 b. An applicatio n t o harmonic analysis 4 9

Notes and reference s 5 1

v

VI CONTENTS

Chapter 5. Grothendieck's Theorem 5 3 a. Preliminaries . Localization techniques . J^-space s 5 3 b. Operators on C( K )-spaces 5 4 c. Operator s on Lj-space s 5 7 d. Cotype 2 spaces and absolutely summing operators 6 2 e. Bes t constants. Krivine's proof 6 4 f. A proof o f G.T . using harmonic analysis 6 8

Notes and reference s 6 9

Chapter 6. Banach Spaces Satisfying Grothendieck's Theorem 7 1 a. G.T . space s 7 1 b. G.T. spaces of cotype 2 7 3 c. Quotient s of L l b y a reflexive subspac e 7 8 d. Bourgain's theorem on L x/H

l 8 3 Notes and reference s 8 6

Chapter 7. Applications of the Volume Ratio Method 8 9 Notes and reference s 9 5

Chapter 8. Banach Lattices 9 7 a. The Banach lattice version of G.T . 9 7 b. Ultraproducts. Factorization throug h an L^-spac e 10 1 c. Loca l unconditional structure . The Gordon-Lewis property 10 4 d. Examples of Banac h spaces without l.u.st . 10 8 e. Finite-dimensiona l space s with extreme l.u.st. constants 11 3 f. G.T . spaces with unconditional basis 11 4 g. Infinite-dimensional Kasi n decompositions 11 4

Notes and reference s 11 8

Chapter 9. C *-Algebras 11 9 a. The noncommutative version of G.T. 11 9 b. Applications 13 0

Notes and reference s 13 2

Chapter 10. Counterexamples to Grothendieck's Conjecture 13 5 a. Outline of th e construction 13 5 b. Extensions of a Banach space 13 7 c. The construction 14 0 d. Particular cases of the conjectures 14 5 e. Som e open problems 14 6

Notes and reference s 14 7

References 149

Introduction

In 195 6 Grothendiec k publishe d a fascinatin g pape r entitle d Resume de la theorie metrique des produits tensoriels topologiques. Thi s paper , whic h i s no w referred t o a s "th e Resume" , ha s ha d a considerabl e influenc e o n th e develop -ment o f Banac h spac e theor y sinc e 1968 . I t containe d a genera l theor y o f tenso r norms o n tenso r product s o f Banac h spaces , describe d severa l operation s t o generate new tensor norms from som e known ones , and studie d th e duality theor y of thes e norms . Bu t th e highligh t o f th e Resum e i s a resul t tha t Grothendiec k called "th e fundamenta l theore m o f th e metri c theor y o f tenso r products " an d which i s no w calle d Grothendieck' s theore m (o r sometime s Grothendieck' s in -equality). Among it s man y consequences , i t implie s tha t ever y bounded operato r from L^ int o L Y factor s throug h L 2. Thi s theore m remaine d practicall y un -noticed unti l 1968 , whe n Lindenstraus s an d Pelczyhsk i revive d i t an d gav e a detailed proo f (cf . [L-P]) . Although ther e ar e now numerous simpl e proofs o f thi s theorem (cf . e.g . Chapte r 5) , it remain s a nontrivia l result .

The aim o f th e present lectur e notes is to describe the contributions made sinc e 1968 i n th e direction s opene d b y th e Resume . Although ou r titl e i s very general , we will limitat e ourselve s t o th e wor k whic h i s directl y relate d t o th e question s raised i n Grothendieck' s paper . The Resum e ends with a list o f si x problems wit h comments o n eac h o f them . Thank s t o th e considerabl e progres s achieve d i n Banach spac e theor y i n th e las t 1 5 years , thes e problem s ar e no w al l solve d (except perhap s fo r th e exac t valu e o f th e Grothendiec k constant) , an d thes e lecture note s wil l include th e various result s which led t o thei r solution . These six problems ar e actuall y al l linke d togethe r an d relate d t o severa l centra l questions . To summariz e simpl y th e content s o f thes e notes , we might sa y tha t the y revolv e around th e followin g questions : When doe s a n operato r u: X -* Y (betwee n tw o Banach spaces ) facto r throug h a Hilber t space ? Fo r whic h space s X, Y doe s thi s happen fo r al l operator s w ? W e wil l examin e th e particula r cas e o f operator s defined o n a Banach lattice , a C*-algebra , o r the disc algebra and H°°.

The topic s tha t w e cove r hav e man y connection s o r application s outsid e Banach spac e theory , an d w e hop e tha t the y wil l hav e eve n mor e i n th e future . With thi s in mind, we have tried t o make this material accessible to nonspecialists , so tha t ou r redactio n i s usuall y quit e detaile d an d self-contained . Fo r th e sam e

vn

Vlll INTRODUCTION

reason, w e have deliberatel y kep t t o a minimum th e use o f th e dualit y theor y vi a the trace , sinc e w e fee l tha t thi s migh t tur n of f th e reader s wh o ar e no t familia r with it . Nevertheless , we urge th e readers wh o want t o go deeper i n th e theor y t o get acquainte d wit h th e principle s o f thi s dualit y (cf . [P I o r Pe4]). We shoul d mention tha t ou r restricte d selectio n ha s lef t ou t severa l importan t topics . W e refer t o [PI ] fo r th e genera l theor y o f operato r ideal s whic h wa s develope d b y Pietsch an d hi s schoo l sinc e the late sixties . The characterization o f L^-space s (o r subspaces o f L p o r subspace s of quotient s o f L p) b y operator theoreti c propertie s is a majo r omission . Fo r this , w e refe r th e reade r t o th e beautifu l pape r o f Kwapieh [Kw3 ] and t o it s references . Also , the factorizatio n theorem s o f Maure y (and th e importan t wor k o f Rosentha l [R2] ) ar e no t include d here ; w e refe r th e reader t o [Ml] . W e d o discuss , however , th e genera l theor y o f typ e an d cotype , but briefl y an d withou t proofs . W e wil l be mainly concerne d her e wit h typ e 2 or cotype 2 . I n general , w e hav e concentrate d o n th e proble m o f factorin g a n operator throug h L 2, an d w e hav e lef t ou t th e natura l extension s fo r th e factorization throug h L p. I n ou r exposition , w e will come acros s mos t o f th e lin e of investigatio n whic h forms th e so-called loca l theory o f Banac h spaces—i.e. , th e study o f Banac h space s by finite-dimensiona l methods . We have trie d t o indicat e in th e references , a s ofte n a s possible , th e ramification s o f thi s currentl y ver y active area .

Let u s no w revie w th e content s o f thes e notes . I n Chapte r 0 , we introduce th e projective an d injectiv e tenso r products an d th e approximation propert y (i n shor t A.P.). Amon g th e si x problem s a t th e en d o f th e Resume , th e firs t an d mos t famous on e wa s th e approximatio n problem : Doe s ever y Banac h spac e posses s the A.P. ? Enfl o [E ] gav e a counterexampl e i n 1972 , whic h opene d a ne w er a i n functional analysis .

In Chapte r 0 , w e hav e insiste d o n th e necessar y distinctio n betwee n nuclea r operators an d element s o f th e projectiv e tenso r product , whic h i s essentia l i n Chapter 10 .

In Chapte r 1 , we present in detail the basic theory of /^-summin g operators an d its firs t application s t o Banach spac e theory: Fo r ever y ^-dimensiona l subspac e E of a spac e X, ther e i s a projectio n P: X - * E suc h tha t ||P| | < yfn an d a n isomorphism T: 1%-> E suc h that | |7 | | \\T~l\\ < yfn.

In §c , we briefly introduc e ^-integra l operator s an d som e rudiments o f dualit y theory, bu t thi s i s no t use d i n th e sequel . W e not e i n passin g tha t th e Radon -Nikodym propert y (whic h is crucial t o compare integra l an d nuclea r operators ) i s not discusse d a t al l here; we refer th e reader t o [D-U] for thi s topic. In Chapte r 2 , we give the Lindenstrauss-Pelczyhsk i criterio n fo r a n operato r t o factor throug h a Hilbert space . This can be viewed a s an applicatio n o f th e Hahn-Banach theore m provided a certain dualit y theore m i s explicited; w e do thi s in §2.b . In Chapte r 3 , we introduc e th e notion s o f typ e an d cotyp e an d prov e Kwapien' s theore m tha t every space of typ e 2 and o f cotype 2 is isomorphic to a Hilbert space . The theor y of typ e an d cotyp e provide s a usefu l scal e t o measur e ho w clos e a give n spac e i s

INTRODUCTION IX

from a Hilber t space . We briefly revie w the main point s of thi s theory in §3.3 (we use onl y th e extrem e cases o f typ e 2 or cotyp e 2 in th e sequel) . In Chapte r 4 , we prove a factorization theore m which links Kwapieh's theore m and Grothendieck' s theorem: I f X* an d Y ar e of cotype 2, then every approximable operato r fro m X into Y factor s throug h a Hilber t space . Thi s resul t play s a crucia l rol e i n th e construction o f Chapte r 10 . As an application , in §4.b, we show that Sido n set s in the dua l o f a compac t Abelia n grou p G ar e characterize d b y th e fac t tha t the y span a cotyp e 2 spac e i n C(G). Thi s generalize s a n earlie r resul t o f Varopoulo s [VI]. I n Chapte r 5 , w e concentrat e o n Grothendieck' s theorem , whic h w e ab -breviate G.T . Chapte r 5 contains a t leas t fou r proof s o f tha t theorem . In §5.a , we briefly introduc e ^-space s (ther e i s more informatio n i n §§8. b an d 8.c) . We ar e mainly concerne d her e with th e case s p = 1 and p = oo . This allow s us t o stat e and prov e G.T. in the framework o f [L-P] : Every operator fro m a n 2£ x spac e into an ££ 2 spac e is 1-summing. Thi s is proved i n §5.c . In §5.b we give the (somewha t dual) formulatio n abou t operator s define d o n a C(AT)-spac e o r o n a n o^-space . We trie d t o give explicitly al l the various forms i n which the theorem can be used , and w e distinguishe d carefull y betwee n th e eas y par t (whic h w e cal l th e "littl e G.T.") an d th e mor e delicat e par t o f thi s theorem . W e firs t giv e a proo f derive d from th e mor e "abstract " resul t o f Chapte r 4 , bu t §5. d contain s anothe r proof , more direct an d o f independent interest .

In §5.3 , we include Krivine' s proo f o f G.T. , whic h give s th e bes t know n uppe r bound fo r th e constant K G. I n problem 3 in th e Resume, Grothendieck aske d fo r the exac t valu e o f variou s constant s (se e 5.3 fo r details) ; thi s is the only proble m which i s no t completel y solve d (bu t o f course , i t i s probably th e leas t importan t one!). In 5.f , w e give a very quick proof o f G.T. , based o n a property o f th e spac e Hl

9 du e to Pelczyhski and Wojtaszczyk . In Chapte r 6 , w e stud y th e Banac h space s satisfyin g G.T. , whic h w e cal l G.T .

spaces. We include several characterizations of thes e spaces, but we insist more on the a prior i smalle r clas s o f G.T . space s o f cotyp e 2 . Th e latte r enjoy s nice r stability properties and includes all the known examples of G.T . spaces. In 6.c, we show tha t i f R i s a hilbertian (or , more generally, a reflexive) subspac e of L l9 the n LY/R i s a G.T. space of cotype 2.

We com e her e t o proble m 5 i n th e Resume . A stronge r formulatio n o f thi s problem wa s give n b y Lindenstraus s an d Pelczyhski , wh o aske d whethe r th e ^ - spaces ar e th e onl y space s satisfyin g G.T . Th e abov e resul t o f 6. c (du e t o Kisliakov an d th e author ) gives a negative answer t o thi s question (an d a fortior i to problem 5 ) since the quotients L x/R ar e never ^-spaces whe n R i s a reflexiv e infinite-dimensional subspac e of L v

It i s rathe r eas y t o giv e concret e example s o f hilbertia n subspace s o f L x: fo r example, th e spa n o f th e Rademache r functions . However , i t i s mor e delicat e t o produce " very large " suc h spaces . Fo r thi s purpose , w e presen t i n Chapte r 7 a method base d o n volum e estimates whic h yield s a n orthogona l decompositio n o f l\n int o tw o part s whic h ar e uniformly (wit h respec t t o n) isomorphi c t o /£ . This

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 o f 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 whethe r 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 o f 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 "wors t 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 o f 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 . I n §7.b , we discuss (withou t 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 tha t 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 wel l 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 grea t 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 . Haageru p 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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