378 13. WEP and LLP13.2. Tensorial characterizations of the LLP and WEP
To get tensorial characterizations of the LLP and WEP, we will need another
striking theorem of Kirchberg.
Theorem 13.2.1 (Kirchberg). For any free group lF and any Hilbert space
Ji, we have
C* (JF) ®max Jlll(Ji) = C* (JF) ® llll(Ji)
canonically.A bit of thought reveals that one only needs to consider the case lF = lF 2
and 1i = C^2. Our proof follows Pisier's beautiful ideas, and requires two
lemmas.
Let En be the n-dimensional operator space in C* (JF n-1) spanned by the
unit Uo = 1 and the standard unitary generators U1, ... , Un-1 of C*(lFn-1).
It is not hard to see that En is canonically isometric to c; (the n-dimensional
C^1 space), or equivalently,
n-1 n-1
II L akUkll = L lakl,
k=O k=O
for all (ak)k,:6 E en. By duality, we have a one-to-one correspondence
between elements z = I:k:6 Uk® Xk E En® llll(C^2 ) and maps
n-1
Tz: Cc:' 3 (ak)k,:6 1--> L akXk E llll(C^2 ).
k=O
Lemma 13.2.2. The above operator space En is canonically completely iso-
metrically isomorphic to the dual operator space c; = (CC:)*, or equivalently,
llzllmin = llTzllcb for every z E En® llll(C^2 ).Proof. Since (Uk)k,:6 E En® C';! is contractive and
z = (idEn ® Tz)((Uk)k,:J) E En® llll(C^2 ),we have llzllmin ~ llTzllcb· To prove the opposite inequality, we give ourselves
contractions ao, ... , an-1 E llll(Ji) and let ak E Mz(llll(1i)) be their unitary
dilations (see the proof of Theorem 13.1.3). It follows that the map cp: En --+
Mz(llll(Ji)) defined by cp(Uk) = a 01 ak, k = 0, ... ,n-1, is c.c. since it extends
to a -homomorphism on C(IB'n-1). Hence, the map 0: En--+ llll(H) defined
by O(Uk) = (aocp(Uk))n = ak, k = 0, ... ,n - 1, is also c.c. Therefore, we
have
ll(idB(7-i) ® Tz)((ak)k,:;6)llmin = 11(0 ® idB(£ 2 ))(z)llmin ~ /lzllmin·
Since the contractive element (ak)k,:6 E llll(Ji) ®C~ was arbitrary, llTzllcb ~
llzllmin and we are done. D