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

TL;DR

This paper constructs a Banach $*$-algebra similar to Cuntz algebras that is purely infinite itself but has ultrapowers that are not simple, revealing new insights into the infiniteness properties of Banach algebras.

Contribution

It introduces a novel Cuntz-like Banach $*$-algebra that is purely infinite but has ultrapowers lacking simplicity, contrasting with known $C^*$-algebra properties.

Findings

Constructed a purely infinite Banach $*$-algebra with non-simple ultrapowers.

Proved the algebra has non-zero traces, distinguishing it from $L^p$-analogues.

Demonstrated combinatorial proof of pure infiniteness.

Abstract

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 ultrapowers are. 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 whose ultrapowers are not even simple, and hence not purely infinite. This algebra is a naturally occurring analogue of the Cuntz algebra, and of the $L^p$-analogues introduced by Phillips. However, our proof of being purely infinite is combinatorial, but direct, and so differs from existing proofs. We show that there are non-zero traces on our algebra, which in particular implies that our algebra is not isomorphic to any of the $L^p$-analogues of the Cuntz algebra.