## Us and them: thoughts the day after

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

## How useful is it that the Gelfand representation is a left adjoint?

(The question won’t get answered in this post, but it has been bugging me sufficiently that I may as well throw the question online, admit my ignorance, and see if anyone has any suggestions or critiques.)

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.)

## Housekeeping

First of all, apologies to the small number of people who have been reading the posts about the “central amenability constant” of a finite group. When I started the sequence of posts, the goal was to force myself past a certain amount of writer’s block, in the hope that this would help to get a preprint written up. Since then there have been some fairly significant changes in my working life — not least a change of jobs and change of continent — and also various other research projects have had to take priority.

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

**a Banach algebra usually looks nothing like an L**The world of Banach algebras can be much stranger and, for me at least, much richer.

^{1}-group algebra.## James Bond will return

… because nothing that makes money will ever be laid to rest. (Exhibit A.)

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…

Well, as usual I’ve not kept up enough with the blog. *Déménagement* has taken priority over the last few weeks. As it happens, while dusting off a suitcase that hasn’t been used for *years*, I found this item inside:

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.

## Removing the Stone of Shame

Following on, in a sense, from the previous post: soon I shall be rid of this turbulent priest, erm, I mean, teaching calculus to 1st year North American students.

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

## Our dream of safety has to disappear

The post title is from W. H. Auden’s Leap Before You Look — full text at this page.

Years ago it was pointed out to me that the rhyme scheme is

abab bbaa baab abba aabb baba

illustrating rather neatly that “4 choose 2 equals 6″. Note also that the last word of each stanza alternates between “leap” and “disappear”, and that there is a kind of “reflectional symmetry” in the order of the stanzas. Specifically, the transposition of a and b has the effect of reversing the order of the 6 4-tuples.

Hmm, maybe I should try this as an example if I get to teach a course introducing people to finite groups…