Extensions and Liftings

There are three important steps in the proof the BDF theorem for essentially normal operators having essential spectrum X ⊆ C. (1) Ext(X) has a neutral element. (2) Ext(X) is a group (i.e., has inverses). (3) Ext(X) depends only on the homotopy class of X. In this lecture we will exhibit the neutral element of Ext(X) and describe the generalization of that result to noncommutative C∗-algebras (without proof). We then show that Ext(X) is a group by the method of [Arv75], using a lifting theorem. The third and most difficult step (homotopy invariance) has not been satisfactorily simplified. An account of the best proof known to date can be found in [Dav96].