Blog of Matthew Daws

An update

An update of sorts. I have finally submitted my long paper "Quantum graphs: different perspectives, homomorphisms and quantum automorphisms".

We have been teaching back in person all year, and really, long may this continue. Given the sort of teaching I do, to the sort of students I teach, in person lectures seem much, much better from a continuing engagement point of view. This year I modified my first year tutorials, making them fully "flipped" and completely de-coupled from any homework. I think this worked well: students seemed more willing to experiment and collaborate when there is no "assessment" (formative or otherwise). Next we need to work on encouraging out of class work.

I wrote before about virtual seminars. I have continued to find travel difficult for a variety of reasons. As such, I have really enjoyed being able to attend various virtual seminars; let me mention the Athens Functional Analysis and Operator Algebra seminar, and the (virtual) Quantum Groups Seminar. It is hard to think that pre-pandemic, I essentially attended no seminars at all. Long may virtual things continue!

Back to in-person seminars

Currently IWOTA2020 is running, a year late and on Zoom. Many of the talks I have attended have been really interesting, but I am not convinced that a big conference like this works well over Zoom. For me, much of the point of large conferences is that I find myself in unusually parallel sessions, learning about things I haven't seen before. I can catch up with old friends and acquaintances, or make new ones, over the coffee breaks. What I've noticed here, and at other large meetings, is that attendance at the parallel sessions often seems rather small. I guess the motivation to go to something a bit random "out of interest" (but maybe, really, because the coffee has run out, and there is not much else to do) doesn't work when video-conferencing.

Anyway, on Monday, I gave an in-person talk at a satellite session, in a small meeting in honour of Garth Dales's retirement. It was slightly weird to be talking in person again. What I more noticed was how much I enjoyed listening to (the other) talks in person. It's so much easier to concentrate, and make good notes, when I'm sat there in person in lecture room. Both Niels and Jared gave nice talks, but it cannot just be down to their skill as speakers that I found the whole event so much more rewarding than Zooming.

Sideways

Another, more frivolous, listen has been Matthew Syed's excellent Radio 4 series Sideways. In particular, Episode 6 made me think of academia. (But all episodes are well worth a listen, IMHO)

In this episode, "A Recipe for Happiness", in the 2nd half, Syed talks about the experiment at Zappos with Holacracy, a rather flat management style (but listen to the episode to really understand what is meant).

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.