Decomposition rank and absorbing extensions of type I algebras

Abstract

We prove the following: Let A and B be separable C * -algebras. Suppose that B is a type I C * -algebra such that (i) B has only infinite dimensional irreducible * -representations, and (ii) B has finite decomposition rank. If 0 → B → C → A → 0 is a unital homogeneous exact sequence with Busby invariant τ , then the extension τ is absorbing. In the case of infinite decomposition rank, we provide a counterexample. Specifically, we construct a unital, homogeneous, split exact sequence of the form 0 → C ( Z ) ⊗ K → C → C → 0 which is not absorbing. In this example, Z is an infinite-dimensional, compact, second countable topological space. This gives a counterexample to the natural infinite-dimensional generalization of the result of Pimsner, Popa and Voiculescu.