1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

90 3. Tensor Products

the extension <.p: JB(.€^2 (r)) --+ JB(.€^2 (r)) given by The Trick takes values in

L(I'). D

It turns out that if A C JB(7t) is an exact C* -algebra, then the inclusion

question we have been considering can fail badly. The proof requires the

easy direction of a difficult theorem.

Lemma 3.6.10.^15 Assume that e: A --+ B is a nuclear map. Then for

every C* -algebra C the map e ®max idc: A ®max C --+ B ®max C factors

through A® C. That is, there exists a c.c.p. map W: A® C--+ B ®max C

such that the diagram

commutes, where A ®max C--+ A® C is the canonical quotient map.

Proof. Let 'Pn: A --+ Mk(n) (C) and 'I/Jn: Mk(n) (C) --+ B be c.c.p. maps

converging to e in the point-norm topology. Due to the fact that there is

a unique C*-norm on Mk(n)(C) 8 C, we get an approximately commuting

diagram

(i®maxidc

A ®max C B ®max C

l ~ I ~n®maxidc

A®C \On®1dc. Mk(n)(C)®C.

Hence we can define a sequence of c.c.p. maps Wn: A® C--+ B ®max C by

Wn =('I/Jn ®max idc) o ('Pn ® idc).

It follows that the algebraic tensor product map e 8 idc: A 8 C --+ B 8 C is
contractive from the spatial norm on A 8 C to the maximal norm on B 8 C
(since Wn(x) --+ e 8 idc(x) for all x E A 8 C) and hence it extends to a
contractive linear map W: A® C --+ B ®max C. Finally, one checks that W
is the point-norm limit of the Wn's, hence is completely positive. D

Proposition 3.6.11. If AC JB(7t) is an exact C*-algebra and C is arbitrary,
then the restriction of II · Jlmax on JB(1t) 8 C is always the spatial norm on
A0C.

Proof. When A is exact, any inclusion AC JB(7t) is nuclear and hence there
is a c.c.p. map^1 ¥: A@C--+ JB(7t)®maxC such that w(a®c) = a®c E JB(7t)8C

(^15) The converse of this lemma is shown in Corollary 3.8.8.