Herrera's
Approximation
Problem
Chapter 16
Paul Halmos introduced the notion of a quasidiagonal operator (on a separa-
ble Hilbert space) over 30 years ago. The purpose of this chapter is to study
approximation properties of such operators and to give a complete solution
to a question of Domingo Herrero. At present there is no operator-theoretic
solution - we will have to rely on the theory of exact C* -algebras.16.1. Description of the problem
Definition 16.1.1. An operator S E IIB(1i) is called block diagonal if there
exist finite-rank projections Pn :S: Pn+l which converge strongly to the iden-
tity and such that [T, Pn] = 0 for all n EN. We denote the set of all such
operators by BI>(1i).
Keep in mind the matrix picture of block diagonal operators. If we write
then the matrix of S with respect to this decomposition is really "block
diagonal" with finite-dimensional blocks.
Definition 16.1.2. An operator T E IIB(1i) is called quasidiagonal if there
exist finite-rank projections Pn :S: Pn+l which converge strongly to the iden-
tity and such that ·11 [T, Pn] II ---+ 0 for all n E N. We denote the set of all such
operators by QI>(1i).
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