1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

110 3. Tensor Products

and let X be the image of X. By definition of the spatial norm, we have an

inclusion

E ® (fin Mk(n) (C)) c IIB('H ® K)

EBnMk(n)(C)

for some (nonseparable) Hilbert space JC, and hence it is easy to see that

llXll =sup ll'Ps ® id(X)ll,

s

where the norm on the right is computed in

Ms(C)@ ( flnMk(n)(C)).

EBnMk(n)(C)

However, M 8 (<C) is @-exact and so we can invert the isomorphism given by

Lemma 3.9.5 to get an isomorphism

Ms(C)@ (flnMk(n)(C))-+ fln(Ms(C)@Mk(n)(C)).

EBnMk(n)(C) EBn(Ms(C)@Mk(n)(C))

Now one must check that the element

n

is a lift of the image of X under the mapping

E@ ( flnMk(n)(C)) -+ Ms(C)@ ( flnMk(n)(C))

EBnMk(n)(C) EBnMk(n)(C)

It follows that

llXll =sup ll'Ps ® id(X)ll

s

fln(Ms(C) @Mk(n)(C))

-+ EBn(Ms(C) @Mk(n)(C))'

=sup (lirnsup lllPs @idk(n)(Xn)ll),
s n->oo
since the norm of (cn)n +EB Cn E (fl Cn)/(EB Cn) is equal to limsup llcnll·
But, <p 8 = <p 8 o 'Pn for all n > s and hence
sup ( lim sup II IPs @ idk(n) (Xn) II) .'S lim sup II IPn @ idk(n) (Xn) II·
s n---+oo n---+oo
Putting this all together, we get
llXll > f3-^1 = limsup lllPn ® idk(n)(Xn)ll 2: llXll
n->oo
and this contradicts @-exactness of A. D