A purely infinite Cuntz-like Banach $*$-algebra with no purely infinite ultrapowers

We continue our investigation, from cite{dh}, of the ring-theoretic infiniteness properties of ultrapowers of Banach algebras, studying in this paper the notion of being purely infinite. It is well known that a $C^*$-algebra is purely infinite if and only if any of its ultrapower is. We find examples of Banach algebras, as algebras of operators on Banach spaces, which do have purely infinite ultrapowers. Our main contribution is the construction of a Cuntz-like Banach $*$-algebra which is purely infinite, but does not have purely infinite ultrapowers. Our proof of being purely infinite is combinatorial, but direct, and so differs from the proof for the Cuntz algebra. We use an indirect method (and not directly computing norm estimates) to show that this algebra does not have purely infinite ultrapowers.