4

PRELIMINARY RESULT S AND BACKGROUN D

(iii) W e will sa y that a Banac h spac e X ha s the approximation propert y (in

short A.P. ) if the identity operato r I d

x

i s approximable. W e say that X ha s th e

A-bounded approximatio n propert y (in short A-B.A.P. ) if Id^ is A-approximable.

Finally, w e say that X ha s the B.A.P. i f i t ha s the A-B.A.P . fo r som e A . The

relation between the A.P. and the injectivity of J can be summarized as follows.

THEOREM

0.3 . Let X be a Banach space. The natural map J: X* ® X into

X* ® X is injective iff Xpossesses the A.P.

We sketch the proof furthe r below .

The duality theory for the tensor product norms is very important .

Let X, Y b e Banach spaces. We denote by @(X X Y) th e space of all bounded

bilinear form s o n X X Y equipped wit h it s natural norm . Clearl y f ( I X Y) i s

isometric t o B(X, 7* ) or , equivalently, t o B(Y, X*). Th e simplest dualit y theo -

rem state s that the dual of X S Ycan b e identified wit h the space f ( IX Y) (or,

equivalently, wit h B(X, 7* ) or B(Y, X*)). Fo r any T i n B(X, Y*) an d any v in

X 8 7 , v = EJJC, . 8 yi9 w e le t

(T,v) = Z{Tx t,y).

Equivalently, i f F i s th e bilinear for m o n X X Y associate d t o T (define d b y

F(x, y) = (Tx, y) fo r x i n X and y i n 7), we let

(F,v) = Y,F(x i,yi).

Clearly, we then hav e

|7»|||r||H

A

,

and it is immediate tha t

| | r | | = s u p { | ( r , u ) | | i ; e * ® 7 , H A l ) .

The linea r for m v - (T, v) = (F,v) has , by density , a uniqu e extensio n t o

X ® 7 wit h th e same norm. We denote it by the same notation .

Conversely, an y element f in (X ® Y)* define s a bilinear for m F on X X Y

by lettin g

F(x, y) =(# , x ® y) V x G ^ V J G 7 .

In conclusion , wit h thi s duality, 3&(X X 7 ) (or B(X, 7*) ) can be identified wit h

the dual of X ® 7

In th e case whe n X an d 7 ar e both f.d. , we have B(X, 7* ) = X* 6 7* , so

that w e can write

(X& 7 ) * = (X* & 7*),

and therefor e (sinc e all the spaces are now reflexive )

(X® 7 ) * = X* i 7*.

We now return t o the case of X* ® X. In that case , we can define a notion of

trace. Le t us denote b y / th e bilinear for m o n X* X X define d b y I(x*,x) =

(x*, x). I n the preceding correspondence , thi s is the bilinear for m associate d t o