Extensions of K by C(x)

An operator A ∈ B(H) whose self-commutator A∗A − AA∗ is compact is called essentially normal. Two operators A ∈ B(H) and B ∈ B(K) are said to be approximately equivalent if they are unitarily equivalent modulo compact operators; more precisely, if there is a unitary operator U : H → K such that B − UAU∗ is compact. This relation is written A ∼ B, whereas the stronger relation of unitary equivalence will be written A ∼= B. Roughly speaking, A ∼= B means that A and B have the same geometric properties, while A ∼ B means that A and B have the same asymptotic properties (see Chapter 3 of [Arv01]). We begin by discussing the classification of essentially normal operators, and its generalization to the computation of Ext(X), originating in work of Brown, Douglas and Fillmore during the mid seventies [BDF77]. In a subsequent lecture we will describe the connection between those results, quasicentral approximate units and the lifting theorem for nuclear C∗-algebras.