The proofs of Theorem 2.2 of K. J. Dykema and M. Rørdam, Purely infinite simple $C^*$-algebras arising from free product constructions}, Canad. J. Math. 50 (1998), 323--341 and of Theorem 3.1 of K. J. Dykema, Purely infinite simple $C^*$-algebras arising from free product constructions, II, Math. Scand. 90 (2002), 73--86 are corrected.

It is shown that simplicity of the crossed product of a unital AH-algebra with slow dimension growth by an endomorphism implies that the algebra is also purely infinite, provided only that the endomorphism leaves no trace state invariant and takes the unit to a full projection.

Let $A$ be a separable amenable purely infinite simple \CA which satisfies the Universal Coefficient Theorem. We prove that $A$ is weakly semiprojective if and only if $K_i(A)$ is a countable direct sum of finitely generated groups ($i=0,1$). Therefore, if $A$ is such a \CA, for any $\ep>0$ and any finite subset ${\mathcal F}\subset A$ there exist $\dt>0$ and a finite subset ${\mathcal G}\subset A$ satisfying the following: for any contractive positive linear map $L: A\to B$ (for any \CA $B$) with $ \|L(ab)-L(a)L(b)\|<\dt$ for $a, b\in {\mathcal G}$ there exists a homomorphism $h\from A\to B$ such that $ \|h(a)-L(a)\|<\ep$ for $a\in {\mathcal F}$.