Blog of Matthew Daws

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.

  • 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.

Read More →


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.

Read More →

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.

Read More →

Open source software

For various reasons relating to 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.

Read More →

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.

Read More →


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.

Read More →


Because of lockdown, and the desire to occasionally get out the house, I have been exploring the local area more closely. The following are some nice resources:

Read More →


I've spent yesterday afternoon and this morning attending the TALMO conference, from the comfort of my home office, via Zoom. The extremely efficient organisers have already got many of the presentations uploaded to YouTube.

Some links which I made during the talks:

  • OBS Virtualcam : OBS is a great (free) screen recorder which allows e.g. composing different inputs (so overlaying a webcam image at an arbitrary place in a screen capture of just one window). The "Virtualcam" is an output which fakes up a new webcam on your system, hence allowing you to pipe the OBS output into any application which uses a webcam.
  • Accessibility talk slides : This has been rumbling away in the background, but an unhappy confluence of a looming deadline just as we add more things online makes it more important. Particularly discussed was Bookdown. Perhaps it is time to move all my lecture notes to a new format. But it's the input of time required, and a new toolchain to learn...
  • Xournal++ (a fork of Xournal into C++ I think leads to the title) is a FOSS PDF annotator that supports LaTeX. Next time I am marking work I will investigate this.
  • There was a question asking about Tablets. Suggestions included Wacom tablets, and those by xp-pen
  • Videos to help students adapt from A-Levels which is especially important this year, as most students will have missed the end of their A-Level studies. There was some talk that the AMSP would be doing something, but I couldn't find much (at a brief glance).
Read More →
Profile image; rendered glass discs
Recent posts