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

0556606 - MÚ 2023 RIV US eng J - Journal Article
Daws, M. - Horváth, Bence
A purely infinite Cuntz-like Banach *-algebra with no purely infinite ultrapowers.
Journal of Functional Analysis. Roč. 283, č. 1 (2022), č. článku 109488. ISSN 0022-1236. E-ISSN 1096-0783
R&D Projects: GA ČR(CZ) GJ19-07129Y
Institutional support: RVO:67985840
Keywords : Banach ⁎-algebra * Cuntz semigroup * purely infinite * ultrapower
OECD category: Pure mathematics
Impact factor: 1.7, year: 2022
Method of publishing: Limited access
https://doi.org/10.1016/j.jfa.2022.109488

We continue our investigation, from [10], 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 Lp-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 Lp-analogues of the Cuntz algebra.
Permanent Link: http://hdl.handle.net/11104/0330776