1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

13.2. Tensorial characterizations of the LLP and WEP 379

Lemma 13.2.3. Let Xi c lBl(Hi) (i = 1, 2) be unital operator subspaces

and let r.p: X1 --+ X2 be a unital complete isometry. Suppose that X 2

is spanned by unitary elements in lBl(Tt'.2). Then, r.p uniquely extends to

a *-homomorphism between the C* -subalgebras C* (Xi) generated by Xi in

lBl(Hi).

Proof. By Arveson's Extension Theorem, r.p extends to a c.c. map from

ll:l\(11'.1) into ll:l\(11'.2), which we still denote by r.p. Since r.p is unital, it has

to be a u.c.p. map. Since r.plx 1 is isometric and X2 is spanned by unitary

elements, X1 is contained in the multiplicative domain of r.p. Hence, r.p is a

*-homomorphism on C*(X1). D

Proof of Theorem 13.2.1. Thanks to Lemma 13.2.3, it suffices to show
that the formal identity map from En ®minlBl(f^2 ) into C (Fn-1) ®maxlBl(f^2 ) is
c.c. for every n (or just n = 3). We give ourselves z = I:k:6 Uk® Xk E En®
lBl(f^2 ) with llzllmin = 1. By Lemma 13.2.2, the corresponding map T 2 : .ec;;;--+
JBl(f^2 ) is c.c. Hence, by the factorization theorem for completely bounded
maps (Theorem B.7), there exist a Hilbert space 1-l, a -homomorphism
7f: .ec;;; --+ lBl ( 1-l) and isometries V, W E lBl ( f^2 , 1-l) such that T 2 (f) = V 7f (f) W
for f E .ec;;;. We may assume that 1{ = f^2 • Then, ak = 7r(bk)V and bk =
7r(bk)W in lBl(f^2 ) are such that Xk = a'kbk for every k and I:k:J a'kak = 1 =
I:k:J b'kbk. It follows that
n-1 n-1
II L Uk® Xkllc(lF 00 )®ma:xlIB(£2) =II L(l ® ak)(Uk ® bk)llmax
k=O k=O
n-1 n-1
~II L(l ® ak)(l ® ak)ll;i!ll L(Uk ® bk)*(Uk ® bk)ll;i;x = 1.
k=O k=O

This shows that the formal identity from En ®min lBl(f^2 ) into C*(F 00 ) ®max
lBl(f^2 ) is contractive. Since lBl(f^2 ) is stable, our formal identity is also c.c. D

Remark 13.2.4. Theorem 13.2.1 and a modification of Theorem 13.1.6 can
be used to give an alternate proof of the fact that C* (F) has the LLP.

The C-algebra lBl(f^2 ) is universal in the sense that it contains all sepa-
rable C -algebras and has the WEP (Definition 3.6. 7), since it is injective.
The full free group C-algebra C(F 00 ) is universal in the sense that it has
the LP and any separable unital C* -algebra is a quotient of it. With these
simple observations we can now establish tensorial characterizations of the
WEP and LLP.

Corollary 13.2.5. For C* -algebras A and B, we have the following:

(1) A ®max B = A® B canonically if A has the LLP and B has the

WEP;