Weak Semiprojectivity for Purely Infinite C-algebras

The first definition of semiprojectivity for C-algebras was given by Effros and Kaminker in the context of noncommutative shape theory ([3]). A more restrictive definition was given by Blackadar in [1]. Loring introduced a third definition, which he termed weak semiprojectivity, in his investigations of stability problems for C-algebras defined by generators and relations ([9]). Recently Neubüser has introduced a slew of variants, the most important being what he called asymptotic semiprojectivity ([10]). Using the authors’ initials to represent the above notions, the implications among them are: B ⇒ N ⇒ EK, L. All versions of semiprojectivity are of the following form: ∗-homorphisms into inductive limit C-algebras can be lifted (in some sense) to a finite stage of the limit (the precise definitions may be found in section 1, and in the references). As a consequence, among the first (and easiest) examples for which semiprojectivity was established are the Cuntz-Krieger algebras. This drew attention to the class of separable, nuclear, simple purely infinite C-algebras, now commonly referred to as Kirchberg algebras ([12]). Kirchberg, and independently Phillips, have shown that in the presence of the universal coefficient theorem, K-theory is a complete invariant for Kirchberg algebras ([6], [11]). Blackadar proved in [2] that for such algebras, finitely generatedK-theory is necessary for semiprojectivity in the sense of [3]. He conjectured that for these algebras finitely generated K-theory is sufficient for semiprojectivity in the sense of [1], and proved this for the case of free K0 and trivial K1. Szymański extended this to the case where rankK1 ≤ rankK0 ([18]), and in [15] semiprojectivity was proved whenever K1 is free. The conjecture remains open in the case that K1 has torsion. The methods used in all work on the conjecture rely upon explicit models for these algebras, constructed from directed graphs. In another direction, Neubüser used abstract methods to show that (for the algebras under consideration) finitely generatedK-theory is equivalent to asymptotic semiprojectivity. In this paper we study weak semiprojectivity for UCT-Kirchberg algebras. We prove that such an algebra is weakly semiprojective if and only if its K-groups are