Introduction

In 1956 Grothendiec k publishe d a fascinatin g pape r entitle d Resume de la

theorie metrique des produits tensoriels topologiques. This 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. It 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 , it 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 are now numerous simpl e proofs o f thi s

theorem (cf . e.g . Chapte r 5) , it remain s a nontrivial result .

The aim o f th e present lecture 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 is 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 Resume ends with a list of si x problems with

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 15 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 notes wil l include th e various results which led t o their solution . These six

problems ar e actuall y al l linked togethe r an d relate d t o severa l central questions .

To summariz e simpl y th e contents 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 does 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 to 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

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