- References
We redefine the pseudometric don Extw(A) by
k
d([cp], ['lj;]) = i~f L JJucp( vi)u* - 'lj;( vi) JJ.
i=l441(Note that this metric is equivalent to the previous one, since A is generated
by the v/s.) As is well known (and a fun exercise), there is an uncountable
family D of infinite subsets of N such that Jex n ,BJ < oo for every ex, ,6 E D
with ex i= ,6. Thus, it suffices to show Jex n ,BJ < oo implies d([cpa], [cp,a]);::: 8.
Suppose by contradiction that Jex n ,BJ < oo and d([cpa], [cp,a]) < 8. Then,
there exists a contraction U: 'Ha ------t 1-i,a such that
k
L JJ(Adu(ui) - uf) + IK('H,a)JJQ(?-i/3) < 8.
i=l
Thus one has
k = II (t uf 0 uf) + IK(Ji~) O lll(Ji~) II
< d + II (t uf 0 uf) + IK(Ha) O lll(Ji~) 11
k
= 8 + limsupsup JI L ui(m) ® ui(n)JJ
m--+oo mEa nE,6 i-._^1
.::::; k,by the coding property, which is absurd. D
17.5. References
As mentioned already, the main result of this chapter is due to Wasser-
mann, building on Voiculescu's counterexample to Herrera's problem. Our
approach, however, follows [30]. For more on Kirchberg's and Haagerup and
Thorbj¢rnsen's counterexamples see [102] and [81], respectively.