Posted on 25th February 2020
Well, not new really, just I forgot to blog about it. "Ring-theoretic (in)finiteness in reduced products of Banach algebras" with Bence Horváth (currently a postdoc at the Czech academy of sciences). Available at arXiv:1912.07108 [math.FA]
We look at ultrapowers and the asymptotic sequence algebra of Banach algebras. There has been some interest recently in using tools from Model Theory (specifically, the recent area of "Continuous model theory") to study such objects for \( C^* \) and von Neumann algebras. One of our research themes is that things do not work so nicely for Banach algebras, and in particular, one often has to get one's hands dirty (and not use Model Theory results) because Banach algebras are not very "metrical" objects, unlike operator algebras. We construct various counter-examples, and also leave open some tantalizing questions about renormings of some rather concrete algebras.
I worked (in a very "bare hands" way) on ultraproducts in my thesis, and shortly afterwards, and it was fun to return to this topic, but to take a slightly more abstract approach. Something we wrote in the introduction is that we wonder if the asymptotic sequence algebra of a Banach algebra could be an interesting source of (counter-)examples for other problems?