1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

3.6. Inclusions and The Trick 89

Definition 3.6.7. A C*-algebra A C JE(H) is said to have Lance's weak

expectation property (WEP) if there exists a u.c.p. map <I>: JE(H) -----t A**

such that <I>(a) =a for all a EA.

A simple application of Arveson's Extension Theorem shows that the

WEP is independent of the choice of faithful representation.

Corollary 3.6.8. A C* -algebra A has the WEP if and only if for every

inclusion A C B and arbitrary C we have a natural inclusion A ®max C c

B®maxC.

Proof. Assume first that Ac JE(H) has the WEP and Ac B. The inclusion

A '-----t JE(H) extends to a c.c.p. map \[!: B -----t JE(H) by Arveson's Extension

Theorem. Composing with <I> gives a map B -----t A** which restricts to the

identity on A and then Proposition 3.6.6 applies. The converse uses The

Trick just as in the previous proposition. This time take B = JE(Hu), the

universal representation of A, and C =(A**)'. D

We opened this section by claiming that inclusions of tensor products
can be tricky and then proceeded to give several instances where they behave
well. Here is an example where inclusions misbehave.

Pr9position 3.6.9. Let r be a discrete group. Then the following are equiv-

alent:

(1) r is amenable;

(2) C{(r) has the WEP;

(3) the natural inclusion 1,: C{(r) '-----t JE(.C^2 (r)) induces an injective ten-

sor product map

L ®max id: C{(r) ®max C{(r) '-----t JE(.C^2 (r)) ®max C{(r).

In particular, the natural map

L ®max id: C{(r) ®max C{(r) -----t lE(.C^2 (r)) ®max C{(r)

has a nontrivial kernel for every nonamenable group.

Proof. (1) ::::} (2) follows from Theorem 2.6.8 and Exercise 2.3.14, while (2)
::::} (3) is immediate from Corollary 3.6.8.
For the final implication we use The Trick to produce a u.c.p. map
: JE(.C^2 (r)) -----t L(r) such that (x) = x for all x E C{(r). We already saw
in the proof of Theorem 2.6.8 that this is enough to imply amenability of r.