Blog of Matthew Daws

Various listening

I attended the BMC in Glasgow (sadly remotely: I actually had a hotel booked for the original 2020 date, before Covid19 overwhelmed the UK) and amongst the plenary talks, I perhaps most enjoyed the plenary talk by Kevin Buzzard, which is now available online. This is a variation on a talk I'd seen before, and Prof Buzzard says a number of things which resonated with me.

Purely Infinite algebras

(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 \( A \).

Counter-examples to Kaplansky Density

I've added a short document to my Mathematical writeups collection. Thanks to Yemon Choi for a conversation which motivated me to write this up, and to Philip Spain who alerted me to the work of Harry Dowson.

It would be interesting to have a natural Banach Algebra example. By this, I mean a naturally occurring example of a dual Banach Algebra \( M \) and a weak\( ^\ast \)-dense subalgebra \( A\subseteq M \) such that there is a unit vector in \( M \) which is not the weak\( ^\ast \)-limit of any bounded net in \( A \).

What do I mean by "natural" here? Some possible examples:

• Let \( E \) be a reflexive Banach space with the approximation property. Then it's known that \( \mathcal B(E) \) is the bidual of \( \mathcal K(E) \). Thus \( \mathcal K(E) \) is weak\( ^\ast \)-dense in \( \mathcal B(E) \), but in this case, we do have an (isometric) version of Kaplansky Density, because biduals satisfy Goldstine's Theorem. I still think this example is a little interesting: if \( E \) had the metric approximation property, given \( T\in\mathcal B(E) \) it is easy to write down a net in \( \mathcal K(E) \) which converges to \( T \) and is bounded by \( \|T\| \). If \( E \) merely has the approximation property then this seems less clear to me, though it is ensured by the general theory. Update: This was a little naive: if \( E \) is reflexive then it goes back to Grothendieck that \( E \) having the AP already implies that \( E \) has the MAP. So really there is no mystery here.

• Let \( G \) be a locally compact group, and consider \( PM_p(G) \) which by definition is the weak\( ^\ast \) closure of \( PF_P(G) \) inside \( \mathcal B(L^p(G)) \). Excepting when \( p=2 \) I do not know if we have Kaplansky Density here, though if \( G \) has a suitable approximation-like property (amenability, or a weakening) then I believe one can show this: I don't know if this is explicitly in the literature.

TileMapBase

Another release of TileMapBase which fix some warnings which had started to be been displayed, because of changing APIs in some libraries which we use. I also had some minor corrections to docstrings. Released on PyPI.

A quick further retrospective. Do I enjoy writing code? Yes, still. Do I enjoying fighting with the tools to get things to work? No, not in the slightest. Below are some notes to self for how I got various libraries installed.

Raspberry Pi and LEDS again

It's that time of the year again, so out come the Raspberry Pi driven LED pixels. Here are some notes to myself about how to set this all up, before I forget again for another 11 months.

Open source software

For various reasons relating to https://github.com/MatthewDaws/AccessibleLaTeX I have ended up down a rabbit-hole of trying to understand Open Source Software Licenses, and related ideas, like Creative Commons. These are some notes for my own memory if nothing else.

Stalled work

Cleaning out my desk, I came across some plans for further work on predictive policing. It now seems rather unlikely I will have time to pursue these (what with a fixed number of hours in a day, and a desire to be a research in Mathematics, at least at the moment). I thought I might as well record the ideas here.

Notes

A couple of new research notes:

• Some notes on inductive limits of Banach spaces and algebras. I don't have a use for this rather esoteric topic, but a couple of textbooks make (slightly) wrong claims, so I wrote up some notes and carefully checked how far we could get things to work.

• A quick proof showing how to get the Kaplansky Density Theorem by using Arens products, and "elementary" (for various values of elementary) \( C^* \)-algebra theory.