428 16. Herrera's Approximation Problem
there. For example, multiplying
by T gives a nondiagonal matrix unit and similar fiddling gives the rest. D
Remark 16.L.l.2. Though we are after explicit examples, the proposition
above shows that starting with any finitely generated QD nonexact C -
algebra (e.g., C(IB'n)), we can construct a counterexample to Herrera's orig-
inal problem by passing to matrices over the given algebra and representing
them on a Hilbert space.
To get concrete examples, we start with a finitely generated residually
finite nonamenable discrete group r. We fix a descending sequence of nor-
mal, finite-index subgroups rk whose intersection is the neutral element. For
many classical groups, like SL( n, Z), these subgroups and. their quotients are
easily described explicitly.
For each k we letbe the representation induced by the left regular representation of r /rk;
then we take their direct sum
1f = E91fk: C*(r) ___, lffi(ffie^2 (r/rk)).
k kTheorem 16.4.3. The algebra 7r(C(r)) is (obviously) RFD, but it is not
exact.^2 Hence sufficiently large matrices over 7r(C(r)) yield a singly gen-
erated nonexact QD C* -algebra, thus concrete counterexamples to Herrero 's
original approximation problem.
Proof. For notational convenience let A= 7r(C*(r)). Note that A has an
amenable trace. T (Theorem 6.2. 7) whose .GNS representation is unitarily
equivalent to the left regular representation of r. (Take any cluster point
of the traces A ---+ lffi( £^2 (I' /rk)) ~ C.) Hence we have a continuous product
map
A 0 A^0 P ___, lffi(£^2 (r))taking A^0 P to the right regular representation of r.
. , ·^2 voiculescu used precisely this cbnstruction but as~umed that' r has property (T). That
7r(C*(r)) is not exact in this more general context was pointed out to.us by Marius Dadarlat.