13.5. Norms on Jlll(.e^2 ) 8 llll(.e^2 ) 389for ( E JC. Under this identification, x ® fj E llll(H ®JC) acts on S2(JC, 1-i) as
S2(JC, 1-i) 3 h r-t xhy* E S2(JC, 1-i).If A is a C* -algebra with a tracial state T, then for any unitary elements
u1, ... , Uk E A, we have
k
II L Ui Q9 Ui"A®maxA = k
i=lsince the functional A ®max A 3 I: Xi® Yi r-t I: T(xiyi) E CC is a continuous
state (see Chapter 6). In particular, for any unitary matrices u1, ... , Uk E
MN(CC), we have
k
II L Ui ® UillMN(C)®MN(iC) = k.
i=l
Definition 13.5.3. Let (u1(n), ... ,uk(n)) E MN(n)(CC)k be a sequence of
k-tuples of unitary matrices. We say the family {(ui(n))i=l, ... ,k : n EN} is
coding if
k
sup{JJ 0 '°' ui(m) ®·Ui(n)JJM N(m) (IC)'°'M 'CJ N(n) (IC): mi= n} < k.
i=l
Theorem 13.5.4 (Voiculescu). There exists a coding family of unitary k-
tuples, for every k 2:: 3.
Proof. To prove the theorem, we use the fact that there exists a residu-
ally finite group r with property (T) (e.g., we can take r = SL(3, Z); see
Theorem 12.1.14).
First we must find an infinite sequence of finite-dimensional irreducible
representations 7rn: r -+ MN(n)(CC) such that 7rn is not equivalent to 7rm
whenever n i= m. Using residual finiteness of r, this is easy. Indeed, if
(r n)nEN is a sequence of finite quotients of r, then the left regular repre-
sentations r n -+ Jill ( .e^2 (r n)) extend to representations r -+ Jill ( .e^2 (r n)) ·. Each
such representation is a direct sum of irreducible representations, and infin-
itely many of these irreducible components must be mutually inequivalent
(since r is infinite). Hence we can find representations 7rn with the desired
properties.
Schur's Lemma (Lemma 17.2.1) implies form i= n the unitary represen-
tation 7rm®7rn on 1-im®Hn does not have a nonzero invariant vector (since, by
Lemma 13.5.2, an invariant vector can be identified with a Hilbert-Schmidt
operator T: 1-in-+ 1-im which intertwines 7rm and 7rn)· Let Sc r be a finite
subset of generators that contains the unit e. (We remark that SL(3, Z) is