I sing myself to sleep

A song from the darkest hour

Secrets I can’t keep

Inside of the day

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To be honest, I’m more annoyed with the *Grauniad* for continuing to indulge him, than with his views…

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Cependant j’étais fière de ce premier travail. Ces journées passées à écrire pour la première fois un texte personnel et qui fut publié (dont l’inimportance m’était peu apparente) m’avaient procuré un grand plaisir. C’est ce même plaisir de produire quelque chose de personnel que je retrouve identique à travers les années : travailler dans mon bureau lorsque tout est calme, taper sur le clavier de mon ordinateur, agrafer les minces liasses de feuilles couvertes de calculs précieux sortant de lâ€™imprimante, mes documents en ordre autour de moi, la corbeille à papiers vite débordante.

The papers that feel like this rather make up for the ones that feel like chores, or “salvage jobs”…

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If a power was to lift him up,

Make him rich, would he admit it was luck?

Or say he’d earned it, claim a state of grace,

Just like the rich in this hateful place?

I’m willing if not able, I am stretched to my limit,

It won’t take very much more to break my mind;

I watch a stable life drift by, it’s out of my range —

This morning paid me back my fear and let me keep the change

A woman drove her Saturn into the black water

Killed herself and her two kids strapped in the back seat

She’d lost her job, and didn’t want her kids to be poor

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Recently, on MathOverflow, I offered the following example of an adjunction that comes up in the theory of commutative unital Banach algebras. Let CHff be the category of compact Hausdorff spaces and continuous maps between them; and let unCBA be the category of unital commutative Banach algebras, with the morphisms being the continuous unital algebra homomorphisms between the objects.

There is a functor C from CHff^{op} to unCBA, defined on objects by taking C(X) to be the usual algebra of continuous **complex**-valued functions on X, and defined on morphisms in the obvious way. Years ago I remembered convincing myself that not only does the functor C have a left adjoint, but one can define/describe the left adjoint as being the functor which assigns to a unital commutative Banach algebra its character space . Here is defined to be the set of characters (=non-zero multiplicative functionals from A to the ground field ), equipped with the relative weak-star topology that this set inherits from the dual Banach space .

What started to nag at me, after mentioning this example on MathOverflow, is that nothing in this description seems specific to the choice of complex scalars; in other words, it looks like one would obtain the same corresponding adjunction if one worked with unital commutative Banach algebras over rather than over . The choice of complex scalars is important because without it one does not get the Gelfand-Mazur theorem, and without that one does not get the fact that all maximal ideals in unital commutative Banach algebras have codimension one, and without that one does not get the following key feature of the Gelfand representation :

if and ${\cal G}_A(a)$ is invertible in , then is invertible in .

So the question arises: just what does one get from knowing the Gelfand representation arises as a left adjoint? What traction does it give us on the well-known examples and theorems in the theory of commutative Banach algebras? (This has been on my mind on and off for several years, because there are various possible generalizations and extensions of the Gelfand representation, either by passing to the noncommutative world or by looking at more general classes of ideals, not just the maximal ones; and I had hoped that the “left adjoint” perspective could be used as a guide when examining which of these versions is going to lead to a good theory. But if the categorical perspective I’ve outlined above can’t lead us to Gelfand-Mazur, then perhaps a rethink is needed.)

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Indeed, the result that I hoped to present in this sequence of blog posts can now be found on the arXiv at

[1410.5134] A gap theorem for the ZL-amenability constant of a finite group

Nevertheless, I still think it may be worthwhile to resume the sequence of posts in the New Year. Rather than serving as a practice run for a preprint, they will instead take the opportunity to be more discursive and explanatory. In particular, I want to try and motivate some of the calculations rather than just stating and proving the theorems, and perhaps include a few more explicit examples.

The other vague project for the New Year is to do some blogging about Banach algebras. Here, the maxims will be: **a Banach algebra usually looks nothing like a C ^{*}-algebra**; and

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On a different note, if I can get some respite from viruses and visa headaches, blogging here may also return. It remains to be seen if I have enough energy and focus to finish off the series of posts on the central amenability constant of a finite group (which nowadays I have tentatively dubbed the ZL-amenability constant). At this rate the paper may actually get finished and submitted before the blog posts, which wasn’t the intention, but is probably the sensible way round to do these things…

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I went back to old haunts in 2011 to collect my MMath, and found that a bookstore I was rather fond of was gone. Not quite Martin Blank finding his old home turned into a convenience store, but it still made me a touch maudlin.

Ah well. *Tempus fugit*, and all that; you can’t cling on to auld lang syne forever, even if marketed nostalgia is one of the staple products of our culture. I still wish that they’d kept more 2nd-hand bookstores and had fewer plastic bars/shops, though.

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(This post brought to you from the Department of Procrastination.)

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