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

30 January, 2015

(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 CHffop 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 \Phi: \hbox{unCBA} \to \hbox{CHff}^{\rm op} which assigns to a unital commutative Banach algebra A its character space \Phi_A. Here \Phi_A is defined to be the set of characters (=non-zero multiplicative functionals from A to the ground field \bf C), equipped with the relative weak-star topology that this set inherits from the dual Banach space A^*.

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 \bf R rather than over \bf C. 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 {\cal G}_A: A \to C(\Phi_A):

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

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

30 December, 2014

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 L1-group algebra. The world of Banach algebras can be much stranger and, for me at least, much richer.

James Bond will return

29 September, 2014
tags:

… 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…

“Memories were meant to fade, Lenny. They were designed that way for a reason.”

1 January, 2014
tags: ,

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:

The happy highways where I went / And cannot come again

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

15 November, 2013

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

10 October, 2013

The post title is from W. H. Auden’s Leap Before You Lookfull 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…

The central amenability constant of a finite group: Part 4 of n

6 September, 2013

Well, that break was longer than intended…

In the last post, we claimed that \displaystyle  {\rm AM}_{\rm Z}(G)>1 for every finite, non-abelian group G. It turns out that the easiest way to prove this goes via a certain minorant for {\rm AM}_{\rm Z}(G) 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

\displaystyle  \inf \{ {\rm AM}_{\rm Z}(G) \colon G \mbox{ finite and non-abelian} \} > 1\;.

1. Recap

Recall that

\displaystyle  {\rm AM}_{\rm Z}(G) = \sum_{C,D\in{\rm Conj}(G)} |C|\ |D| \left\vert \sum_{\phi\in {\rm Irr}(G)} \frac{1}{|G|^2} \phi(e)^2\phi(C)\phi(D) \right\vert \;.

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

\displaystyle  \overline{D} = \{ x^{-1} \colon x\in D \}

also belongs to Conj(G). Moreover, the map {D \mapsto \overline{D}} is an involution, in particular is bijective. Therefore, since {\phi(\overline{D})=\overline{\phi(D)}} for every character {\phi}, we obtain

\displaystyle  {\rm AM}_{\rm Z}(G) = \sum_{C,D\in{\rm Conj}(G)} |C|\ |D| \left\vert \sum_{\phi\in {\rm Irr}(G)} \frac{1}{|G|^2} \phi(e)^2\phi(C)\overline{\phi(D)} \right\vert \;.

We already saw this idea, in a special case, when we looked at {{\rm AM}_{\rm Z}(G)} 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.

2. A remark on normalized versus unnormalized counting measure

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 {\ell^1}-norm. That is, we are looking at {L^1(G,\lambda)} where {\lambda} 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 {L^1(G,\mu)} where G is a compact group and {\mu} denotes uniform probability measure on G.

However, there is no serious conflict. For if G is a finite group, let A denote {\ell^1(G)} equipped with counting measure {\lambda} and equipped with convolution using {\lambda}, and let B denote {\ell^1(G)} equipped with uniform probability measure {\mu} and equipped with convolution using {\mu}. 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 {{\rm AM}_{\rm Z}(G)} coincides with the formula in [ASS09] for the amenability constant of {L^1(G,\mu)}.

3. A minorant for {{\rm AM}_{\rm Z}(G)}

At a naive level (but not a completely facile one) we might say that the difficulty in getting non-trivial lower bounds on {{\rm AM}_{\rm Z}(G)} 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 {{\rm AM}_{\rm Z}(G)}, 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 {{\rm AM}_{\rm Z}(G)} and not its precise value, we are free to discard terms indexed by particular C and D. Thus, as observed in [ASS09], {{\rm AM}_{\rm Z}(G)} is bounded below by the following quantity

\displaystyle  \begin{aligned} \alpha(G) & := \sum_{C\in{\rm Conj}(G)} |C|^2 \left\vert \sum_{\phi\in {\rm Irr}(G)} \frac{1}{|G|^2} \phi(e)^2\phi(C)\overline{\phi(C)} \right\vert \\ & = |G|^{-2} \sum_{C\in{\rm Conj}(G)} \sum_{\phi\in {\rm Irr}(G)} \phi(e)^2 |\phi(C)|^2 |C|^2 \end{aligned} \ \ \ \ \ (1)

(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 {\rm AM}_{\rm Z}(G) > 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) Let G be a finite, non-abelian group, and let

\displaystyle  \begin{aligned} s & =\min \{ |C| \colon C\in {\rm Conj}(G), |C|\neq 1 \} \\ & \equiv \min \{ |\mbox{conj. class of } x | \colon x \in G\setminus Z(G) \}.\end{aligned}

Then

\displaystyle  \alpha(G) \geq 1 + (s^2-s)|G|^{-2} > 1 \;.

Proof: Compare the formula (1) which defines \alpha(G) with

\displaystyle  |G|^{-2} \sum_{C\in{\rm Conj}(G)} \sum_{\phi\in {\rm Irr}(G)} \phi(e)^2 |\phi(C)|^2 |C| \ \ \ \ \ (2)

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

\displaystyle  |G|^{-2} \sum_{\phi\in {\rm Irr}(G)} \phi(e)^2\sum_{C\in{\rm Conj}(G)} |\phi(C)|^2 |C| = |G|^{-1} \sum_{\phi\in {\rm Irr}(G)} \phi(e)^2 = 1.

Hence

\displaystyle  \alpha(G)-1 = \sum_{C\in{\rm Conj}(G)} \sum_{\phi\in {\rm Irr}(G)} \phi(e)^2 |\phi(C)|^2 (|C|^2-|C|) \;.

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 {\phi(C)=0}) but we can identify at least one strictly positive term. Namely, let {C_0} be a conjugacy class of size s, and consider the trivial character {\varepsilon} which takes the value 1 everywhere. Then

\displaystyle  \varepsilon(e)^2 |\varepsilon(C_0)|^2 (|C_0|^2-|C_0|) = s^2-s \geq 2,

which gives us the lower bound that was claimed. \hfill\Box

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 \alpha(G) over all finite non-abelian groups G strictly greater than 1?

Nevertheless, as mentioned in the first post of this series, we can do better when it comes to {{\rm AM}_{\rm Z}(G)}, 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) Let K be a compact group, let E be a finite subset of Irr(K), and let {f=\sum_{\phi\in E} \phi(e)\phi \in L^1(K)}. (The orthogonality relations for irreducible characters imply that {f} is a central idempotent in {L^1(K)}, and all central idempotents in {L^1(K)} arise this way.) If {\Vert f\Vert_1 > 1}, then {\Vert f \Vert_1 \geq 301/300}.

Now let G be a finite, non-abelian group. Since {\rm AM}_{\rm Z}(G)\geq \alpha(G), Proposition~1 immediately implies that {{\rm AM}_{\rm Z}(G) > 1}. Now {{\rm AM}_{\rm Z}(G)=\Vert \Delta_G \Vert}, where {\Delta_G} is a central idempotent in {L^1(G\times G)}. Applying Rider’s theorem to {\Delta_G} we deduce, as in [ASS09], that {{\rm AM}_{\rm Z}(G)\geq 301/300}.

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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