Posted on 10th May 2021
(Written much later than it should have been, and back-dated). "A purely infinite Cuntz-like Banach ∗-algebra with no purely infinite ultrapowers" with Bence Horváth. Available at arXiv:2104.14989 [math.FA].
This is a follow-on paper from our previous work. We previously look at various notions of "ring-theoretic infiniteness" for Banach algebras, and how these were (or were not) stable under the ultrapower/product constructions. The one (main) notion we didn't consider was that of being "purely infinite", which is well studied for C*-algebras, and, as it turns out, has also been studied for general (simple) algebras. We take the definition that a unital (Banach) algebra \( A \) is purely infinite if each non-zero \( a\in A \) there are \( b,c\in A \) with \( bac=1 \). Again, this property is stable under the ultrapower construction exactly when we have "sufficient norm control": in this case, that we can choose such \( b,c \) with \( \|b\| \|c\| \) uniformly bounded, as \( a \) varies over the unit sphere of \( A \).
While purely-infinite C*-algebras are stable under ultrapowers, we conjectured that this wouldn't be true for Banach algebras. The issue then is to construct a counter-example, but this seems a little hopeless, as being purely infinite is a rather "global" property: simply playing with weighted semigroup algebras seems likely to destroy the original algebra being purely infinite. In short, we needed a good supply of purely infinite Banach algebras to play with.
After noodling around for some time, and getting nowhere, Bence suddenly noticed that a simple lemma gave us everything we needed: if \( A \) has purely infinite ultrapowers, then (continuous) homomorphisms out of \( A \) are bounded below. We were already working with the semigroup algebra of the "Cuntz monoid" (quotiented by the one-dimensional ideal generated by the semigroup zero) which has been studied before in the Banach Algebras literature. Unfortunately, some calculations of mine turned out to be nonsense, but motivated by now having some hope of a result, we noticed that if we further quotient the semigroup algebra by the relation \( 1 = s_1s_1^\ast + s_2s_2^\ast \), closely mirroring the Cuntz algebra.
Some combinatorial arguments now establish this Banach algebra is purely infinite, but there is a natural representation on \( \ell^1 \), leading to a homomorphism to \( \mathcal B(\ell^1) \), which is not bounded below. Thus, we have our counter-example. At this point, we realised that the closure of the image on \( \mathcal B(\ell^1) \) was an example of the \( L^p \)-operator algebras studied by Phillips (and coauthors and students), see for example arXiv:1201.4196 [math.FA]. However, our algebra seems not to have been studied before, which is perhaps surprising. Further, we could not see any non-trivial implications between our results and the results of Phillips (for example, that "our" algebra is purely infinite seems to say nothing about whether the closure of a homomorphic image should be purely infinite, and vice versa).