13.6. References 391It follows that C* ( {Vi : i = 1, ... , k}) c M / K cannot have an amenable
tracial state (since, by Theorem 6.2.7, such a trace T would give a state on
C* ( {Vi : i = 1, ... , k}) 0 C* ( {Vi : i = 1, ... , k}) defined by x 0 y 1--'t T ( xy*))
and hence cannot be quasidiagonal (Proposition 7.1.16).We now present Pisier's proof of Theorem 13.5.1.Proof of Theorem 13.5.1. Let {(ui(n))i=l,. .. ,k E MN(n)(C)k : n EN} be
a coding family of unitary k-tuples. Let (Jn: C*(IB'k) -t MN(n)(<C'.) be the
*-homomorphism given by (Jn(Ui) = ui(n) for every i = 1, ... , k, where
the Ui's are the free generators of C*(IB'k)· Taking a subsequence, we may
assume that the sequence trN(n)((Jn(x)) converges for every x E C*(IB'k)· For
each j = 1, 2, let
Mj = IJ MN(2n+j)(C) and u~j) = (ui(2n+ j))nEN E Mj.
nEN
Let Aj = C*({u~j): i = 1, ... ,k}) C Mj. Fix a free ultrafilter won N and
consider the GNS representations 7rj of the tracial states Tj on Mj, which are
given by Tj((xn)n) = liIDw trN(2n+j)(xn). Then, by assumption, we have anisomorphism from 7r1(A1) onto 7r2(A2) sending 7r1(u~^1 )) to 7r2(u~^2 )). We set
N = 7rj(Aj)^11 and Vi= 7rj(u~j)) EN. Since there exists a (trace-preserving)
conditional expectation from 7rj(Mj)^11 onto 7rj(Aj)^11 , there exist u.c.p. maps
l.{Jj: Mj -t N such that 1.fJj(u~j)) =Vi for j = 1, 2.
By the coding assumption, we have
k - k
II L u~^1 ) 0 u~^2 )11Mi®M 2 =sup II L ui(2m + 1) 0 ui(2n + 2)11 < k.
i=l m,n i=l
On the other hand, since N has a tracial state, we have
k - k
II L u~l) 0 u~2
) llM1®maxM2 2 II L Vi 0 VillN®maxN = k.
i=l i=lIt follows that M10maxM2 # M1 @M2. Since Mj <-+ lffi(.€^2 ) in such a way that
Mj is the range of a conditional expectation, this proves the assertion. 0
- References
Most of the results in this chapter are due to Kirchberg [102]. Notable
exceptions are the proof of Theorem 13.2.1, which is taken from [150], and
Junge and Pisier's Theorem 13.5.1, which is taken from [93, 153]. See [134]
for more information on the QWEP conjecture.