# 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!

• For a reference (of a similar version) check Rordams book 'classification of nuclear C∗-algebras', chapter 4. There is a similar version with a proof, where A is assumed to be simple, separable, not stably finite and B is simple and not of type I. This indicates that your stated version is probably wrong if you drop these conditions. – Sabrina Gemsa Jun 10 '18 at 23:16
• I guess one can try the following (or a suitible modification of it): Try the calkin algebra $A=B(\ell^2)/K(\ell^2)$ together with the suspension of the complex numbers, $B=S\mathbb{C}$, as an counterexample. – Sabrina Gemsa Jun 10 '18 at 23:17

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.

• Hello Jamie! Welcome to Mathoverflow! – Ulrich Pennig Jun 29 '18 at 11:29
• Hi, Jamie. It seems that the result you mentioned at first appears in a recent paper of Kirchberg-Sierakowski on arXiv: Filling families and strong pure infiniteness, arXiv:1503.08519. – Targaryen Jul 2 '18 at 2:57
• Ah fantastic, this paper had (unreasonably) slipped my attention! I also see that the example I gave above is exactly Example 6.3 of said paper. – Jamie Gabe Jul 4 '18 at 7:50