1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

426 16. Herrera's Approximation Problem

Proof. By Exercise 2.3.10 we may assume that T E Band(H). However,

every banded operator can be identified with an element in .e= (Z) ><1 Z, since

this algebra is precisely the norm closure of the banded operators on R^2 (Z)

(see Proposition 5.1.3) and we can always identify 1-i with R^2 (N) c R^2 (Z)

in such a way that T lands inside .e= (Z) ><1 Z. Since exactness passes to

subalgebras and .e=(z) ><1 Z is nuclear (Theorem 4.2.4), the proof is complete.

D

Here's the answer to Herrera's problem:

Theorem 16.3.3. BVbdd(ri) = QV(ri) n Band(H). In other words, if there

exists a banded sequence Sn such that llT - Snll ----+ 0 and a block diagonal

sequence Un such that llT-Unll ----+ 0, then there exists a sequence Xn which

is simultaneously banded and block diagonal such that llT -Xnll ----+ 0.

Proof. We already observed that BVbdd(ri) c Q'D(H)nBand(H) and hence

the same inclusion with norm closures is immediate. Assume now that T E

QV(H)nBand(H). Since Tis quasidiagonal, one easily checks that C*(T) c

IBS(H) is a quasidiagonal set of operators. The previous lemma implies that

C*(T) is also exact. Hence, by Theorem 16.2.6, there exist operators Tn such

that llT-Tn II ----+ 0 and each of the C* -algebras C* (Tn) is finite-dimensional.

However, standard representation theory of finite-dimensional C* -algebras

shows that.each Tn is both block diagonal and banded, and this completes

the proof. D

16.4. Counterexamples

The first counterexamples to Herrera's original question were discovered

by Szarek ([181]). His proof was nonconstructive, however, and in [189]

Voiculescu gave the first explicit examples of quasidiagonal operators which

don't generate exact C* -algebras. To do this, he constructed finitely gen-

erated QD C* -algebras which aren't exact. Passing to a single operator

requires the following trick:

Proposition 16.4.1. Let A = C* (b1, ... , bn) be a unital C* -algebra gen-

erated by n self-adjoint elements. Then Mn (A) is a singly generated C* -

algebra.

Proof. We only sketch the ingredients required. The first step is to show
that we may assume that each bi 2 0, bi is invertible and the spectra <Y(br)
are pairwise disjoint. (Hint: replace bi with bi+AilA for appropriate positive
constants Ai· Invertibility implies the algebra generated by bi+AilA contains
bi-)