2

PRELIMINARY RESULT S AND BACKGROUN D

In general , w e refe r th e reade r t o th e book s o f Lindenstraus s an d Tzafrir i

[L-Tl, L-T2 ] fo r unexplaine d terminolog y an d fo r mor e backgroun d i n Banac h

space theory .

b. A n introductio n t o tenso r products . Th e approximatio n property . Nuclea r

operators. Let X, 7 b e Banach spaces. Consider the linear tenso r product X ® Y.

Each element v in X ® Y ha s a representatio n

n

(0.1) v = £ x

t

® y

t

fo r x

t

e X, y

t

e 7 .

I

Moreover, w e may clearly regard v also as a bilinear for m o n X * X 7 * mappin g

an element (x*, *) in X* X 7 * intoEx*(;c z)^*(j/).

We ca n als o associat e t o o a finit e ran k operato r v: X* - 7 whic h i s

a( A"*,

Ar)-continuous;

this operator i ; is defined b y

vx* e x*, £(**) = 53 **(*/).yl..

The injective tenso r product nor m | | | |

v

i s defined o n ! 0 7 a s follows:

(0.2) H v = H k ^ )

= sup{X*(x,)*(^)|x *

G

*** ' ^ *

G B

r*)-

This nor m coincide s wit h th e operato r nor m o f u o r wit h th e nor m o f u a s a

bounded bilinea r for m o n I * X 7* .

The projective norm | | | |

A

is defined b y

(0-3) lkl U = inf|i:i|x,||||^||}

where the infimum run s over all possible representations (0.1). Clearly, we have

(0.4) I M I V ^ H I A .

Let a b e a norm o n X ® 7 . We denote by X ®

a

7 th e space X ® 7 equippe d

with th e nor m a . W e cal l a "reasonable " i f th e bilinea r map s (x , y) - x S y

from X X 7 int o X 0 7 an d (*', y') - + JC ' 8 / fro m X * X 7 * int o (X ®

a

7) *

are both o f nor m 1. This implies a(x ® y) = \\x\\ \\y\\. With thi s terminology, it

is eas y t o chec k tha t th e norm s | | | |

A

an d | | | |

v

are , respectively , th e larges t an d

the smallest reasonable norm o n l® 7 .

The completio n o f A" ® 7 wit h respec t t o ||||

v

(resp . | | || A) i s calle d th e

injective (resp . projective ) tenso r produc t o f X an d 7 an d i s denote d b y X ® 7

(resp. X i 7). I t i s easy t o se e that an y elemen t v i n X 8 7 ca n b e represente d

as a series v = Ef

3

x

n

® yn wit h x

n

i n X , „ in 7 suc h tha t E||xJ | ||jJ| oo , and

||y||

A

is the infimum o f EJ° ||jtJ| || j„|| over all such representations.

The spac e X i 7 ca n be described easily ; by (0.2) it can be identified wit h th e

closure i n B(X*, 7 ) o f al l th e o(X*, X)-continuou s operator s o f finit e rank .

However, i t i s more delicate to describe X E 7 a s a space of operators . By (0.4),

we have a norm decreasin g map /: X ® Y -* X k 7 . But , in general , this map i s

not injective.