Pure infiniteness of tensor product $C^\ast$-algebras I found the following theorem (Proposition 4.5) in the paper of E. Kirchberg and M. Rørdam: Non-simple purely infinite $C^\ast$-algebras. Amer. J. Math. 122(3) (2000), 637-666 (MR1759891, jstor, author's homepage).
Let $A$ be a unital, simple, separable, nuclear, purely infinite $C^\ast$-algebra, and let $B$ be any $C^\ast$-algebra. Then $A\otimes B$ is purely infinite.
My question is that can we drop the condition of nuclearity or separability  of $A$ but the same conclusion still hold? Is there some known results?
I have searched in the literature but nothing founded. 
Thank you very much for all helps!
 A: Kirchberg announced quite a lot of years ago that $A\otimes_{\min{}} B$ is strongly purely infinite if $A$ is strongly purely infinite and $B$ is exact. Unfortunately, I don't think this result has ever been published.
Different results of the same nature are contained in [Blanchard, Etienne; Kirchberg, Eberhard Non-simple purely infinite C∗-algebras: the Hausdorff case. J. Funct. Anal. 207 (2004), no. 2, 461–513.]. For instance, Corollary 3.9 gives 8 different criteria for when $A\otimes_{\min{}} B$ is locally purely infinite, e.g. when $A$ is locally purely infinite and $B$ is exact.
In particular, if $A = B(\ell^2)/K(\ell^2)$ and $B = S\mathbb C$ (as suggested in the comments by S. Ge), then $A\otimes_{\min{}} B$ is locally purely infinite. By Proposition 5.1 and Theorem 5.8 from the before mentioned paper it follows that $A\otimes_{\min{}} B$ is strongly purely infinite.
EDIT: The following is an example of separable, simple, unital, exact $C^\ast$-algebras $A$ and $B$ such that $A$ is purely infinite, but $A\otimes_{\max{}} B$ is not (locally) purely infinite. Hence it is important to work with minimal tensor products if one wants to preserve pure infiniteness.
Let $F_2$ be the free group on two generators, and let $C^\ast_r(F_2)$ denote the reduced group $C^\ast$-algebra. Let $D$ be a separable, simple, unital, nuclear, finite $C^\ast$-algebra which is not stably finite (the existence of such is due to Rørdam [Rørdam, Mikael A simple C∗-algebra with a finite and an infinite projection. Acta Math. 191 (2003), no. 1, 109–142.]). Then $A = D \otimes C^\ast_r(F_2)$ and $B = C^\ast_r(F_2)$ does the trick.
By well-known results $A$ and $B$ are separable, simple, unital and exact. By Theorem 4.1.10 in Rørdam's book "Classification of nuclear, simple C∗-algebras" it follows that $A$ is purely infinite. By a result of Akemann and Ostrand, there is a tensor product $C^\ast_r(F_2) \otimes_\nu C^\ast_r(F_2)$ which contains a two-sided closed ideal isomorphic to the compact operators $\mathcal K(\ell^2(F_2))$. In particular, $C^\ast_r(F_2) \otimes_\nu C^\ast_r(F_2)$ contains a hereditary $C^\ast$-algebra isomorphic to $\mathbb C$. Hence
$$
A\otimes_{\max{}} B \cong D \otimes (C^\ast_r(F_2)\otimes_{\max{}} C^\ast_r(F_2))
$$
has a quotient isomorphic to $D \otimes (C^\ast_r(F_2) \otimes_\nu C^\ast_r(F_2))$ which contains a hereditary $C^\ast$-subalgebra isomorphic to $D \otimes \mathbb C = D$.
As (local) pure infiniteness passes to quotients and hereditary $C^\ast$-subalgebras, and as $D$ is not (locally) purely infinite, it follows that $A\otimes_{\max{}} B$ cannot be (locally) purely infinite.
