1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

422 16. Herrera's Approximation Problem

Remark 16.1.3. Note that Proposition 7.2.3 shows T E QV(H) if and only

if {T} is a quasidiagonal set in the sense of Definition 7.2.1. From the local

formulation of quasidiagonal sets, it is evident that QV(H) is a norm closed

set of operators (which contains block diagonal operators).

We can now reformulate Theorem 7.5.1, which is really due to Halmos,

as follows.

Theorem 16.1.4. If T E QV(H), then for every c > 0 there exists S E

BV(H) such that T-SE JK(H) and llT -Sii < c. In particular,

QV(H) = BV(H),

where the closure is taken in norm.

For reasons we will soon explain, the following subset of QV(H) is im-

portant.

Definition 16.1.5. An operator S E IIB(H) is called block diagonal with

bounded blocks if there exist finite-rank projections Pn ::; Pn+l which con-

verge strongly to the identity, such that [T, Pn] = 0 for all n E N and

suprank(Pn - Pn-1) < oo.

n

We denote the set of all such operators by BVbdd(H).

Again, thinking of block matrices and the decomposition

this terminology should be clear.
The study of various examples suggested that the norm closures of
BV(H) and BVbdd(H) may coincide (i.e., for a while, all the known· ex-
amples of quasidiagonal operators could be shown to lie in the norm closure
of BVbdd(H)). So, Herrero asked ifthis was the case:
Is every quasidiagonal operator in the norm closure of BVbdd(H)?
It turns out that the answer is "no" and we will describe some explicit
counterexamples in the last section of this chapter. However, knowing that
QV(H) properly contains the norm closure of BVbdd(H), the original ques-
tion naturally evolved into Herrera's approximation problem, which asks:

What is the norm closure of BVbdd(H)?

The answer to this question is found in Theorem 16.3.3.