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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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…

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In the last post, we claimed that for every finite, non-abelian group **G**. It turns out that the easiest way to prove this goes via a certain minorant for which we will work with in some subsequent posts. In this post, we’ll introduce this minorant, give an explicit lower bound, and then briefly indicate how it allows us to show the stronger result that

Recall that

We can rewrite this in a cosmetic but suggestive way. Observe that the inversion map on **G**, which sends each element to its inverse, maps conjugacy classes to conjugacy classes. It follows that for each D in Conj(**G**), the set

also belongs to Conj(**G**). Moreover, the map is an involution, in particular is bijective. Therefore, since for every character , we obtain

We already saw this idea, in a special case, when we looked at for abelian groups. There, the point of this small change was that it made the expression look more like an inner product, so that one could apply Schur orthogonality relations; a similar idea was applied in a recent paper of Alaghmandan, Samei and myself (arXiv 1302.1929) to handle certain groups which are close to the abelian case in some sense.

First, I need to clear up an issue of normalization conventions, which I omitted to deal with before. In our series of posts, we have always been working on the complex group algebra equipped with the -norm. That is, we are looking at where denotes counting measure on the finite set **G**.

On the other hand, the paper of Azimifard–Samei–Spronk (henceforth referred to as [ASS09]), where the amenability constant of the centre of the group algebra was first studied, considers where **G** is a compact group and denotes uniform probability measure on **G**.

However, there is no serious conflict. For if **G** is a finite group, let **A** denote equipped with counting measure and equipped with convolution **using **, and let **B** denote equipped with uniform probability measure and equipped with convolution **using **. Then a direct calculation shows that the obvious isometric rescaling map from **A** to **B** is in fact an isomorphism of Banach algebras. In particular, **A** and **B** have the same amenability constant. Thus, our formula from coincides with the formula in [ASS09] for the amenability constant of .

At a naive level (but not a completely facile one) we might say that the difficulty in getting non-trivial lower bounds on is due to the fact that one takes the modulus of a sum of different terms, inside which there might be significant cancellation. Indeed, this is exactly what happens in the case of an abelian group: see the previous post for details.

One situation where we can avoid cancellation is where the terms in the sum are all non-negative, so that the modulus is just the sum itself. Looking at the revised formula for , we see that this happens whenever *C=D* (it may also happen for some other choices of *C* and *D*, but let us ignore that for now). Moreover, if we only want a lower bound on and not its precise value, we are free to discard terms indexed by particular *C* and *D*. Thus, as observed in [ASS09], is bounded below by the following quantity

(The paper [ASS09] does not give this quantity a specific symbol, but in subsequent posts it will appear frequently enough that some extra notation seems warranted.)

In the previous post, we claimed that if **G** is a non-abelian finite group then we have > 1. We can now give a sharper statement. (The calculation in [ASS09] does not give the explicit bound that we do, but it is implicit in their work.)

Proposition 1 (Azimifard–Samei–Spronk, 2009)LetGbe a finite, non-abelian group, and letThen

*Proof:* Compare the formula (1) which defines with

Rearranging the sum and using the Schur row and column orthogonality relations, we see that (2) is equal to

Hence

Now all of the terms on the right hand side are non-negative. Some of them may be zero (for instance, whenever *C* consists of just a single point, or whenver ) but we can identify at least one strictly positive term. Namely, let be a conjugacy class of size *s*, and consider the trivial character which takes the value 1 everywhere. Then

which gives us the lower bound that was claimed.

Note that our lower bound “gets worse” as **G** gets bigger. Indeed, I believe the following question is still open.

Question.Is the infimum of over all finite non-abelian groupsGstrictly greater than 1?

Nevertheless, as mentioned in the first post of this series, we can do better when it comes to , which is the original quantity of interest. This was done in [ASS09] by appealing to a hard result of D. A. Rider, which tells us that the norms of central idempotents have “a gap at 1″.

Theorem 2 (Rider, 1973)LetKbe a compact group, letEbe a finite subset of Irr(K), and let . (The orthogonality relations for irreducible characters imply that is a central idempotent in , and all central idempotents in arise this way.) If , then .

Now let **G** be a finite, non-abelian group. Since , Proposition~1 immediately implies that . Now , where is a central idempotent in . Applying Rider’s theorem to we deduce, as in [ASS09], that .

Rider’s proof is rather long and technical and we will not present the details here. The constant 301/300 is somewhat arbitrary, resulting from choices made in chains of estimates, and can be improved slightly by repeating Rider’s arguments with more nit-picking. However, it seems that a significant improvement in the constant would require new ideas.

In ~~the next~~ a future post, we will see that with a more careful use of the Schur orthogonality relations, one can improve the lower bound in Proposition~1 to a constant that does not depend on |**G**|, provided that **G** has trivial centre. To do this we will need a new ingredient, not available in [ASS09], which ensures that a group which has an irreducible character of “surprisingly large” degree cannot have any small conjugacy classes except for elements of the centre.

**Edited 2013-06-17:** corrected some typos/omissions.

**Edited 2014-12-30:** revised rash promise.

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Given a finite group **G**, denotes the usual complex group algebra: we think of it as the vector space equipped with a suitable multiplication. This has a canonical basis as a vector space, indexed by group elements: we denote the basis vector corresponding to an element *x* of **G** by . Thus for any function , we have .

(Aside: this is not really the correct “natural” way to think of the group algebra if one generalizes from finite groups to infinite groups; one has to be more careful about whether one is thinking “covariantly or contravariantly”. is naturally a contravariant object as **G** varies, but the group algebra should be covariant as **G** varies. However, our approach allows us to view characters on **G** as elements of the group algebra, which is a very convenient elision.)

The centre of , henceforth denoted by , is commutative and spanned by its minimal idempotents, which are all of the form

for some irreducible character . Moreover, is a bijection between the set of irreducible characters and the set of minimal idempotents in .

We define

and, equipping with the natural -norm, defined by

we define to be . Explicitly, if we use the convention that the value of a class function on any element of a conjugacy class **C** is denoted by , we have

the formula stated in the first post of this series.

Remark 1

As I am writing these things up, it occurs to me that “philosophically speaking”, perhaps one should regard as an element of the group algebra , whereG^{op}denotes the group whose underlying set is that ofGbut equipped with the reverse multiplication. It is easily checked that a function on is central as an element of if and only if it is central as an element of the algebra , so we can get away with the definition chosen here. Nevertheless, I have a suspicion that the picture is somehow the “right” one to adopt, if one wants to put the study of into a wider algebraic context.

is a non-zero idempotent in a Banach algebra, so it follows from submultiplicativity of the norm that . When do we have equality?

Theorem 2 (Azimifard–Samei–Spronk)if and only ifGis abelian.

The proof of necessity (that is, the “only if” direction) will go in the next post. In the remainder of this post, I will give two proofs of sufficiency (that is, the “if” direction).

In the paper of Azimifard–Samei–Spronk (MR 2490229; see also arXiv 0805.3685) where I first learned of , this direction is glossed over quickly, since it follows from more general facts in the theory of amenable Banach algebras. I will return later, in Section 2.2, to an exposition of how this works for the case in hand. First, let us see how we can approach the problem more directly.

Suppose **G** is abelian, and let . Then **G** has exactly *n* irreducible characters, all of which are linear (i.e. one-dimensional representations, a.k.a. multiplicative functionals). Denoting these characters by , we have

so that

This sum can be evaluated explicitly using some Fourier analysis — or, in the present context, the Schur *column* orthogonality relations. To make this a bit more transparent, recall that for all characters and all *y* in **G**. Hence by a change of variables in the previous equation, we get

For a fixed element *x* in **G**, the *n*-tuple is a column in the character table of **G**. We know by general character theory for finite groups that distinct columns of the character table, viewed as column vectors with complex entries, are orthogonal with respect to the standard inner product. Hence most terms in the expression above vanish, and we are left with

which equals , since each takes values in . This completes the proof.

The following argument is an expanded version of the one that is outlined, or alluded to, in the paper of Azimifard–Samei–Spronk. It is part of the folklore in Banach algebras — for given values of “folk” — but really the argument goes back to the study of “separable algebras” in the sense of ring theory.

Lemma 3LetAbe an associative, commutative algebra, with identity element1. Let be the linear map defined by . Then there is at most one element_{A}min that simultaneously satisfies =1and for all_{A}ainA.

*Proof:* Let us first omit the assumption that **A** is commutative, and work merely with an associative algebra that has an identity.

Define the following multiplication on :

Then is an associative algebra — the so-called enveloping algebra of **A**. If *m* satisfies the conditions mentioned in the lemma, then

and so, by taking linear combinations, for every *w* in . If *n* is another element of satisfying the conditions of the lemma, we therefore have *n**m*=*m*, and by symmetry, *m**n*=*n*.

Now we use the assumption that **A** is commutative. From this assumption, we see that is also commutative. Therefore

as required.

Now let **G** be a finite group and let **A**= . Because **A** is spanned by its minimal idempotents , and because minimal idempotents in a commutative algebra are mutually orthogonal, satisfies the two conditions mentioned in Lemma 3. On the other hand, if **G** is abelian, consider

Clearly =**1 _{A}**, and a direct calculation shows that for all

as required.

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