Continuing a theme from earlier posts, of finding myself too drained/busy to blog about my recent work but also feeling that in this day and age one must develop the habit of announcing/recording one’s activity…

Bence Horváth, Niels Laustsen and I have recently had the following paper accepted by the Transactions of the American Mathematical Society:

## Approximately multiplicative maps between algebras of bounded operators on Banach spaces

Coauthors: B. Horváth, N. J. Laustsen

We show that for any separable reflexive Banach space X and a large class of Banach spaces E, including those with a subsymmetric shrinking basis but also all spaces Lp for 1 ≤ p ≤ ∞, every bounded linear map ℬ(E) → ℬ(X) which is approximately multiplicative is necessarily close in the operator norm to some bounded homomorphism ℬ(E) → ℬ(X). That is, the pair (ℬ(E), ℬ(X)) has the AMNM property in the sense of Johnson (J. London Math. Soc. 1988). Previously this was only known for E=X= lp with 1 < p < ∞; even for those cases, we improve on the previous methods and obtain better constants in various estimates. A crucial role in our approach is played by a new result, motivated by cohomological techniques, which establishes AMNM properties relative to an amenable subalgebra; this generalizes a theorem of Johnson (op cit.).

The main headline results mentioned in the abstract were obtained by Bence during his PhD studies at Lancaster with Niels and myself. The paper itself, however, takes a slightly different perspective from the original proofs that Bence worked out, in that a large portion of the paper is devoted to a framework of approximate splitting for an approximate cochain complex, which we use to prove an AMNM result relative to an amenable subalgebra. This is an extrapolation from a sketch given by Barry Johnson in his 1988 paper mentioned above, but there he chose to do the main calculation by hand rather than setting up approximate cochain complexes explicitly. My coauthors and I felt that having this machinery written out explicitly would be useful for future researchers.

Strictly speaking, Johnson’s sketch and result apply to AMNM problems where the domain algebra is amenable; our “relative version” is influenced by ideas concerning normalization of (Hochschild) cocycles with respect to an amenable subalgebra, a theme that has been well known to specialists and used in work of many authors (Johnson-Kadison-Ringrose; Christensen-Sinclair-Smith; Lykova; Grønbæk; …), and which I learned long ago from Sinclair-Smith’s book on Hochschild Cohomology of von Neumann Algebras.

The paper had an unusually long gestation time, for which I probably bear most of the blame. Fortunately, my coauthors have been forgiving and the refereeing time was relatively short.

Finally, credit (or blame) for the title of the blog post must go to Mahya Ghandehari and Michael White who independently pointed out to me some years ago that calling something AMNM elicits a Pavlovian response.

tags:

(All excerpts provided thanks to Project Gutenberg.)

But by and large the animals enjoyed these celebrations. They found it comforting to be reminded that, after all, they were truly their own masters and that the work they did was for their own benefit. So that, what with the songs, the processions, Squealer’s lists of figures, the thunder of the gun, the crowing of the cockerel, and the fluttering of the flag, they were able to forget that their bellies were empty, at least part of the time.

It was only his appearance that was a little altered; his hide was less shiny than it had used to be, and his great haunches seemed to have shrunken. The others said, “Boxer will pick up when the spring grass comes on”; but the spring came and Boxer grew no fatter. Sometimes on the slope leading to the top of the quarry, when he braced his muscles against the weight of some vast boulder, it seemed that nothing kept him on his feet except the will to continue. At such times his lips were seen to form the words, “I will work harder”; he had no voice left. Once again Clover and Benjamin warned him to take care of his health, but Boxer paid no attention. His twelfth birthday was approaching. He did not care what happened so long as a good store of stone was accumulated before he went on pension.

Napoleon himself appeared at the meeting on the following Sunday morning and pronounced a short oration in Boxer’s honour. It had not been possible, he said, to bring back their lamented comrade’s remains for interment on the farm, but he had ordered a large wreath to be made from the laurels in the farmhouse garden and sent down to be placed on Boxer’s grave. And in a few days’ time the pigs intended to hold a memorial banquet in Boxer’s honour. Napoleon ended his speech with a reminder of Boxer’s two favourite maxims, “I will work harder” and “Comrade Napoleon is always right”–maxims, he said, which every animal would do well to adopt as his own.

The farm was more prosperous now, and better organised: it had even been enlarged by two fields which had been bought from Mr. Pilkington. The windmill had been successfully completed at last, and the farm possessed a threshing machine and a hay elevator of its own, and various new buildings had been added to it. Whymper had bought himself a dogcart. The windmill, however, had not after all been used for generating electrical power. It was used for milling corn, and brought in a handsome money profit. The animals were hard at work building yet another windmill; when that one was finished, so it was said, the dynamos would be installed. But the luxuries of which Snowball had once taught the animals to dream, the stalls with electric light and hot and cold water, and the three-day week, were no longer talked about. Napoleon had denounced such ideas as contrary to the spirit of Animalism. The truest happiness, he said, lay in working hard and living frugally.

Somehow it seemed as though the farm had grown richer without making the animals themselves any richer — except, of course, for the pigs and the dogs. Perhaps this was partly because there were so many pigs and so many dogs. It was not that these creatures did not work, after their fashion. There was, as Squealer was never tired of explaining, endless work in the supervision and organisation of the farm.

An uproar of voices was coming from the farmhouse. They rushed back and looked through the window again. Yes, a violent quarrel was in progress. There were shoutings, bangings on the table, sharp suspicious glances, furious denials. The source of the trouble appeared to be that Napoleon and Mr. Pilkington had each played an ace of spades simultaneously.

Five submissions in 2020, five publications in 2021. Not sure what can be inferred from a somewhat stochastic process, but at least I no longer hear the wolves at the door.

26. Y. Choi. A gap theorem for the ZL-amenability constant of a finite group. Int. J. Group Th. 5 (2016) no. 4, 27–46.

27. Y. Choi. Realization of compact spaces as cb-Helson sets. Ann. Funct. Anal. 7 (2016), no. 1, 158–169.

28. Y. Choi. Triviality of the generalized Lau product associated to a Banach algebra homomorphism. Bull. Austral. Math. Soc. 94 (2016), no. 2, 286–289.

29. Y. Choi, M. Ghandehari, H. L. Pham. Stability of characters and filters for weighted semilattices. Semigroup Forum 102 (2021), no. 1, 86–103.

30. Y. Choi, M. Ghandehari, H. L. Pham. Unavoidable subprojections in union-closed set systems of infinite breadth. European J. Combin. 94 (2021), article 103311 (17 pages).

31. Y. Choi. Constructing alternating 2-cocycles on Fourier algebras. Adv. Math. 385 (2021) article 107747 (28 pages).

32. Y. Choi, M. Ghandehari. Dual convolution for the affine group of the real line. Complex Anal. Oper. Theory 15 (2021), no. 4, article 76 (32 pages).

33. Y. Choi. A short proof that B(L1) is not amenable. Proc. Roy. Soc. Edinburgh Sect. A 151 (2021), no. 6, 1758–1767.

This is the one year anniversary of this blogpost, announcing an arXiv posting of an article which had not yet been submitted. The arXiv posting was born from a (justified) fear that if I didn’t get some work I had done in 2020 written up and posted to the arXiv, it would just sit on my TO DO list for 2021 and beyond…

In that post, I grumbled that:

In more easy-going times, professionally speaking, I would have preferred to sit on this and use the summer of 2021 to push the techniques in the paper to their limits. But, well, these days one has to keep the wolf from the door.

As it turned out, during March and April 2021, some new ideas emerged as a result of the teaching I had been doing, alluded to here. (This was not so much a case of finding inspiration as seeking escape/relief!) I’ll say more about the new material below.

In addition, I realised during the summer of 2021 that some technical issues which I had overlooked or brushed under the carpet, concerning the unitary dual of the Cartesian square of certain groups, needed to be treated more carefully: this is related to the phenomenon/issue, discussed in various places on Kevin Buzzard’s blog, that as professional mathematicians we get increasingly comfortable – but also too cavalier – about assuming that two things which “surely must be isomorphic in a canonical way” are actually equal.

So, after a couple of months dealing with other demands, and several spells of tearing my hair out over various “proofs by reference to earlier results, or results in other books”, I finally managed to put together a proper revised version of the preprint, posted to arXiv just before Christmas.

## An explicit minorant for the amenability constant of the Fourier algebra

For a locally compact group G, let AM(A(G)) denote the amenability constant of the Fourier algebra of G. We show that AM(A(G)) ≥ 3/2 for every non-abelian G, extending a result of Johnson (JLMS, 1994) who obtained this for finite non-abelian groups. This lower bound, which is known to be best possible, improves on results by previous authors and answers a question raised by Runde (PAMS, 2006).

To do this we study a minorant for AM(G), related to the anti-diagonal in G×G, which was implicitly used in Runde’s work but hitherto not studied in depth. Our main novelty is an explicit formula for this minorant when G is a countable virtually abelian group, in terms of the Plancherel measure for G. As further applications, we characterize those non-abelian groups where the minorant attains its minimal value, and present some examples to support the conjecture that the minorant always coincides with the amenability constant.

## Some remarks on the new material

Although I took the opportunity to do a lot of minor rewrites to the exposition, the most significant change in v2 compared to v1, mathematically speaking, is that the first version merely stopped at showing that AD(Γ) ≥ 3/2 for every non-abelian Γ, while the second version includes a new section that characterizes those Γ for which equality occurs: they are exactly those groups in which the centre has index 4. Such groups also have the property that the derived subgroup has order 2, and hence in various ways they can be thought of as the non-abelian groups that are closest to being abelian.

Originally I worked this out for the finite case (see this blogpost for a self-contained exposition), since this is the setting where we know that the minorant AD actually coincides with the amenability constant of the Fourier algebra. However, since I am hopeful that the two invariants agree for infinite groups as well, it seemed worthwhile to work out as much as possible for AD, for future use/interest.

The proof of the general case is technically more involved than the finite case, since we can no longer use arguments based on counting conjugacy classes. One theme, which didn’t quite survive into the final write-up, but did play an important heuristic role, is that certain statements about the centre and particular centralizers are countably determined, i.e. if they are true for all countable subgroups of Γ then they are true for Γ itself. This was important because the main formula in the paper for AD(Γ) was only proved under a countability assumption; it seems likely that the assumption could be removed, but one would have to reinvent or recheck a lot of material on direct integrals of group representations in order to ensure that the relevant technical machinery still remains valid.

The new version of the paper also includes a new section with some questions raised by this work (the most obvious, and the most important, being whether AD is always equal to the amenability constant of the Fourier algebra). One question which might be worth mentioning here for those interested in finite group theory: if G is a finite group and AD(G) ≤ 2, does it follow that maxdeg(G) ≤2? (This condition on degrees is clearly sufficient to ensure AD(G) ≤ 2; and dihedral groups show that we can get as close to 2 from below as we wish.) A related question, which I didn’t include explicitly in the paper: what is the infimum of AD(G) over all finite groups G which have maxdeg ≥ 3?

tags:

with all the will in the world
diving for dear life
when we could be diving for pearls

Alternative title: “an analyst does some finite group theory, latest in an irregular and infrequent series”. Or: when life gives you lemons, get your engineers to invent a combustible lemon …

Shortly after posting the preprint mentioned in this blogpost to the arXiv, at the tail end of 2020, I started to find minor errors or omissions in the exposition; none of them affect the validity of the main results, but it did mean that some major revisions were needed. Unfortunately, the demands of teaching and assessment and quality assurance (the joys of UKHE) left me with little time and even less energy to focus on doing so.

Instead, since I was teaching a first course on groups (and rings), I found myself playing around with character theory of finite groups to keep myself entertained, and it was during one of these spells when I realized how a result in that paper could be sharpened for finite groups. Subsequently, I worked out a way to obtain the sharper result for all virtually abelian groups, but the proof becomes more technical and has more moving parts, because one does not have access to the same counting arguments that are possible for finite groups.

The proof for the general case is being written up, and will be added to the revised version of the arXiv preprint. But it seemed worth writing up the simpler argument for the finite case, for two reasons: increased forgetfulness as I get older; and increased sentimental attachment as I get older. So here it is. Probably it should have been split over several posts, but then that would have greatly decreased the chances of ever finishing it.

For a finite group G, let Irr(G) be the set of irreducible characters of G (throughout, we are working over the complex numbers, so no modular representation theory to see here). If $\phi\in {\rm Irr}(G)$ then we write $d_\phi=\phi(e)$ for the degree of φ, or equivalently, the dimension of any (irreducible) representation of which φ is the trace. A fundamental result in the classical character theory of finite groups is that $|G| = \sum_{\phi\in {\rm Irr}(G)} (d_\phi)^2$.

Consider the following quantity, which emerged in work of Johnson (J. London Math. Soc. 1994) on the (non-)amenability of Fourier algebras of compact groups, but which in the case of finite groups can be defined directly:

$\displaystyle{\rm AD}(G) := |G|^{-1} \sum_{\phi\in {\rm Irr}(G)} (d_\phi)^3$

Remark 1. The notation AD stands for “antidiagonal”, and does not appear in Johnson’s original paper. The reason for this notation/terminology comes from the 2020 work mentioned above, where AD(G) is defined in a more abstract way — but to keep this blog post more focused, I will not go into this here.

It is my belief that this numerical invariant of G deserves further study. It is fairly obvious from the definition above that ${\rm AD}(G\times H)={\rm AD}(G)\, {\rm AD}(H)$. What is much less obvious, to me at least, is that whenever H is a subgroup of G, we have AD(H) ≤ AD(G). Indeed, I do not know of a direct proof using the definition above, given that the representation theory of a subgroup of G can be radically different from that of G itself; the only proof I know goes via the “abstract definition” that is alluded to above.

So what can we say about AD(G)? Let us start with some results that can be found in Johnson’s 1994 paper. First of all, since $d_\phi \geq 1$ for all φ,

$\displaystyle {\rm AD}(G) = \frac{1}{|G|} \sum_{\phi\in {\rm Irr}(G)} (d_\phi)^3 \geq \frac{1}{|G|} \sum_{\phi\in {\rm Irr}(G)} (d_\phi)^2 = 1$

and this inequality is strict unless $d_\phi=1$ for all $\phi\in {\rm Irr}(G)$. That is:

Proposition 2.
For any finite group $G$, we have ${\rm AD}(G)\geq 1$. Equality holds if and only if $G$ is abelian.

This raises a natural question: if G is a non-abelian finite group, how small can $AD(G)$ be?

On the face of it, is conceivable that one could find non-abelian finite groups $G_n$ such that $AD(G_n) \searrow 1$. But inspecting the argument used to prove Proposition 2, one sees that for $AD(G)$ to be close to 1, there must be a high proportion of 1-dimensional characters among the elments of Irr(G). Now every 1-dimensional character factors through the abelianization $G\to G_{ab}$; more precisely, the 1-dimensional characters of G correspond to the group homomorphisms $G_{ab}\to {\mathbb T}$, and hence by Fourier analysis for finite abelian groups, there are exactly $|G_{ab}|$ of these characters. Since $G_{ab}$ is (isomorphic to) the quotient of G by its commutator subgroup [G.G], and since [G,G] has size at least 2 when G is non-abelian, we see that

$\displaystyle | \{ \phi \in {\rm Irr}(G) \colon d_\phi= 1\} | = \frac{|G|}{|[G,G]|} \leq \frac{|G|}{2}$

and from here, it is a short step to the following result.

Theorem 3. If $G$ is a non-abelian finite group, then ${\rm AD}(G) \geq 3/2$.

Before explaining the proof of the theorem, we note the following corollary.

Corollary 4. If $H_1, \dots, H_n$ are finite non-abelian groups, then ${\rm AD}(H_1\times \dots \times H_n) \geq (3/2)^n$.

In particular, by taking powers of some fixed class-2 nilpotent group, we see that there are class-2 nilpotent groups with arbitrarily large AD-constant.

The proof of Theorem 3. Let $\Omega_n$ denote the set of irreducible characters of G which have degree n, so that

$\displaystyle 1 = \frac{1}{|G|} \sum_{n\geq 1} n^2 |\Omega_n| \hbox{ and } {\rm AD}(G) = \frac{1}{|G|} \sum_{n\geq 1} n^3 |\Omega_n|$

Thus

and rearranging gives ${\rm AD}(G) \geq 2 - |G|^{-1}|\Omega|$. But since $G$ is non-abelian, we know from the remarks before the theorem that $|G|^{-1}|\Omega_1|\leq 1/2$, and the result follows.

So far, all of this is in Johnson’s original paper. However, he stops short of characterizing those non-abelian G for which the extremal value AD(G)=3/2 is attained. Inspecting the proof of Theorem 3, one sees that in order for AD(G) to equal 3/2, the following conditions are both necessary and sufficient:

• Equality must hold in (*);
• $|G|^{-1} |\Omega_1| = 1/2$.

In turn, these are equivalent to the following pair of necessary and sufficient conditions:

Second set of conditions:

It turns out that these conditions can be replaced with one that is purely group-theoretic, i.e. one that does not make any reference to characters.

Theorem 5. Let G be a finite group. TFAE:

2. G/Z(G) is isomorphic to the Klein-four group.

The last condition may be phrased as: G is a non-trivial central extension of $C_2\times C_2$. It should therefore be possible to classify all possible G using the methods of group cohomology, but I haven’t made serious efforts to look up the necessary details.

Remark 6. Instinctively, one feels that $2)\implies 1)$ should be easier than $1\implies 2$, because 2) specifies some structural property of G and 1) is a statement about some numerical invariant of G defined in terms of its representation theory. However, during my original investigations in March/April 2021, it was the implication $1)\implies 2)$ which came first, because as we will see it follows quite easily if one takes for granted certain basic facts about the character table of a finite group. The converse implication is conceptually easy — since Z(G) has small index in G, the commutators in G must have a restricted form — but somehow writing out the proof involves more nitpicking over cases than I expected.

#### Proving the implication $1)\implies 2)$

We start from the “second set of conditions” stated above, and use them to show that $|G:Z(G)|=4$. If we can show this, then G/Z(G) is a group of order 4; and up to isomorphism the only groups of order 4 are $C_4$ and the Klein-four group. But now we have a general group-theoretic fact: if H is any group (not necessarily finite), N is a subgroup of Z(H), and H/N is cyclic, then H is abelian. (Students who I was teaching in the first half of 2021 may recognize this from the mock exam paper!) In particular, in our setting $G/Z(G)$ cannot be cyclic, so it has to be isomorphic to the Klein-four group.

Let us now show that $|G:Z(G)|=4$. The idea here is quite natural if one has ever spent time playing with character tables; note that by the given assumptions on G, all irreducible characters have degree 1 or 2, and we have a great deal of control on the number of degree 1 characters. To be precise, using the notation above, we know that $|{\rm Irr}(G)| = |\Omega_1| + |\Omega_2|$ and we also know that $|G| = |\Omega_1| + 4|\Omega_2|$. Since | [G,G] | =2, the order of $G_{ab}$ is half the order of G, and by Fourier/Pontryagin duality this means $|\Omega_1| = \frac{1}{2}|G|$. Substituting this into the previous equations we obtain

$\displaystyle |{\rm Irr}(G)| = |\Omega_1| + \frac{1}{4}( |G| - |\Omega_1|) = \frac{5}{8}|G| \;.$

On the other hand: let us try to count conjugacy classes in G. In any group H, there is an injection from each conjugacy class into the set of commutators of H, so in particular the size of every conjugacy class is bounded above by the order of the commutator subgroup. Since we are assuming that |[G,G]|=2, it follows that each conjugacy class of G is either a singleton (i.e. an element of the centre) or has size exactly 2. Writing $k_2$ for the number of conjugacy classes of size 2, we have $|G|= |Z(G)|+ 2k_2$ and

$\displaystyle |{\rm Conj}(G)| = |Z(G)|+k_2 = |Z(G)| + \frac{1}{2}(|G|-|Z(G)|) = \frac{1}{2}|G| +\frac{1}{2}|Z(G)| \;.$

But the character table is a square! That is, $|{\rm Irr}(G)|=|{\rm Conj}(G)|$ (one of the most notorious “unnatural bijections” in algebra). We therefore have

$\displaystyle \frac{5}{8}|G| = \frac{1}{2}|G| +\frac{1}{2}|Z(G)|$

and rearranging gives $|G|= 4 |Z(G)|$ as required.$\hfill \Box$

#### Proving the implication $2)\implies 1)$

It suffices to show that 2 implies both of the “second set of conditions“. Pick a non-central element $x_0\in G$, and let $H$ be the subgroup of G generated by $x_0$ and Z(G). Then H is contained in the centralizer $Z_G(x_0)$, which is a proper subgroup of G since $x_0$ is non-central, and so $|G:H|\geq |G:Z_G(x_0)| \geq 2$. On the other hand, Z(G) is a proper subgroup of H since $x_0\in H$, and so $|H:Z(G)|\geq 2$. Hence

$\displaystyle 4 = |G:Z(G)| = |G:H|\, |H:Z(G)| \geq 2 |H:Z(G)| \geq 4.$

Since equality must hold throughout, we see that $|G:H|=2$. Now $H$ is abelian, so all its irreducible characters have degree 1. If $\phi\in {\rm Irr}(G)$, then by considering ${\phi\vert}_H$ and invoking Frobenius reciprocity, we see that φ must occur as a summand of some induced character ${\rm Ind}^G_H \chi$ where $\chi \in {\rm Irr}(H)$. Since H is abelian $d_\chi=1$, and so $d_\phi \leq |G:H|d_\chi =2$. This establishes the first of the two conditions.

It remains to show that $| [ G,G] | =2$. What follows is not the original proof that I came up with — see Remark 7 below — but it preserves more symmetry, at the expense of being less direct.

Let $q:G\to G/Z(G)$ be the quotient map. The Klein-four group has the following properties:

• each non-identity element has order 2 (which by a favourite exercise of those teaching group theory, implies the group is commutative);
• any two distinct non-identity elements generate the whole group.

Now partition $G$ into four distinct cosets of $(G)$, labelled as $Z(G), aZ(G), bZ(G), cZ(G)$. For convenience, let $x\sim y$ denote the equivalence relation “x and y belong to the same coset of Z(G)” (equivalently, $q(x)=q(y)$). Then the properties of the Klein-four group listed above imply:

• $a\sim a^{-1} \;,\; b\sim b^{-1} \;,\; c\sim c^{1}$;
• $ab\sim c\sim ba \;,\; bc \sim a \sim cb \;,\; ca \sim b \sim ac \;.$

The equivalence relation respects multiplication (since q is a homomorphism) and so we also have $b\sim a^{-1}c$, etc.

The key point: a little thought (or exercise) shows that if $x\sim x'$ and $y\sim y'$ then $[x,y]=[x',y']$. (Here it is important that we are quotienting by Z(G) and not by some arbitrary normal subgroup.)

Since $a\sim a^{-1}$ and $b\sim ac$, we have

$\displaystyle [a,b] = [a^{-1},ac] = a^{-1}(ac) a (ac)^{-1} = [c,a]$;

and since $a\sim cb\sim cb^{-1}$, we also have

$\displaystyle [a,b] = [cb^{-1},b] = (cb^{-1})b(bc^{-1})b^{-1}=[c,b]$

Everything said thus far remains invariant under a permutation of the symbols $a,b,c$, since this just corresponds to a relabelling of the non-trivial cosets of Z(G). Therefore, starting from the identities $[a,b]=[c,a]=[c,b]$, we obtain

$\displaystyle [b,c] = [a,b]=[a,c] \quad\hbox{and}\quad [c,a] = [b,c]=[b,a].$

Thus all six expresions $[a,b], [b,c],[c,a],[b,a], [c,b], [a,c]$ are equal to the same element of G, which we denote by $z_0$. Since every element of G is equivalent to one of $\{e,a,b,c\}$, we have shown that $[x,y] \in \{e,z_0\}$ for all $x,y\in G$.

Note that $z_0\neq e$. For since $G\neq Z(G)$, there exist $x,y\in G$ with $xy\neq yx$. But $x$ is equivalent to exactly one element of the set $T:=\{a,b,c\}$ and $y$ is equivalent to another element of $T$, so $[x,y]=z_0$.

Moreover,

$\displaystyle [a,b] = [b,a] =[b^{-1},a^{-1}] = ([a,b])^{-1}$

which shows that $z_0$ is an involution. Thus $\{e,z_0\}\subseteq [G,G] \subseteq \langle z_0 \rangle = \{e,z_0\}$ and we have shown that $|[G,G]=2$ as required. $\hfill \Box$

Remark 7. The proof just given avoided “breaking symmetry”, in the sense that we did not privilege any of the non-trivial cosets of Z(G) over any of the others; equivalently, when considering G/Z(G) we did not pick two specific generators. However, it did rely on some ad hoc trickery using the equivalence relation $\sim$ to show that all non-trivial commutators take the same value. In this context, it should be admitted that the argument above is not the one I first came up with when first proving $2)\implies 1)$ in Theorem 5. The original argument went as follows: fix two noncommuting elements $a_0$ and $b_0$ in G, and note that their images in G/Z(G) commute and generate the whole group, so that G can be written as a disjoint union $Z(G) \cup a_0Z(G) \cup b_0Z(G) \cup a_0b_0Z(G)$ with $b_0a_0Z(G)=a_0b_0Z(G)$. Writing $z_0 = [a_0,b_0]$ one has $z_0b_0 = a_0 b_0 (a_0)^{-1}$; squaring both sides, and using centrality of $z_0$ and $b_0^2$, we obtain

$\displaystyle (z_0)^2(b_0)^2 = (z_0b_0)^2 = a_0 (b_0)^2 (a_0)^{-1}=(b_0)^2$

Thus $z_0$ is an involution. I then did a case by case analysis of the various commutators $[a_0,b_0], [b_0,a_0b_0]=[b_0,b_0a_0]$, etc. and used manipulations similar to those above to show that all non-trivial commutators were equal to $[a_0,b_0]$. Compared with the proof above, this approach feels slightly more “hacky”, but it does seem to suggest more naturally why one might hope to reduced all the commutators to (expressions involving) $[a_0,b_0]$.

Just when I thought I was out, they pull me back in…

New version of 1301.4295 has just been posted to the arXiv. Ah, those halcyon days (as seen through rose-tinted spectacles) where I thought had a chance of keeping up with the (non-)amenability of F.

In the blog posts from last year, I was only mentioning the newly posted preprints which were single-authored, but I neglected to mention three co-authored papers which were written up and submitted during 2020. Originally my plan was to wait until I had some further commentary or explanation to offer before posting about these three papers, but given the long-standing writer’s block on this blog, it seems more constructive to put up some “belated announcements” which can be referred back to, albeit not in any proper order.

The following paper was written up during the summer of 2020, submitted in September 2020, accepted in March 2021, and (after some unfortunate issues with the production process) has just been published online.

## Dual convolution for the affine group of the real line

Coauthor: M. Ghandehari

The Fourier algebra of the affine group of the real line has a natural identification, as a Banach space, with the space of trace-class operators on $L^2({\mathbb R}^\times, dt/ |t|)$. In this paper we study the “dual convolution product” of trace-class operators that corresponds to pointwise product in the Fourier algebra. Answering a question raised in work of Eymard and Terp, we provide an intrinsic description of this operation which does not rely on the identification with the Fourier algebra, and obtain a similar result for the connected component of this affine group. In both cases we construct explicit derivations on the corresponding Banach algebras, verifying the derivation identity directly without requiring the inverse Fourier transform. We also initiate the study of the analogous Banach algebra structure for trace-class operators on $L^p({\mathbb R}^\times, dt/ |t|)$ for $p\in (1,2)\cup(2,\infty)$.

Published online as Complex Anal. Oper. Theory 15 (2021), no. 4, article 76.

### A little extra context

Our adoption of the phrase “dual convolution” is due to an extremely helpful referee report on an older paper of ours; when we wrote the paper we directly studied the multiplication operation on $L^1({\mathbb R}, {\mathcal S}_1({\sf H}))$ corresponding to pointwise product in the Fourier algebra of the real Heisenberg group, but it was the referee who pointed out to us a paper of Stinespring and the terminology “dual convolution”.

The original motivation for seeking an explicit formula for “dual convolution” for ${\mathbb R}\rtimes{\mathbb R}^\times$ comes from an old derailed project (the one alluded to in the introduction and conclusion of the “alternating 2-cocycles on A(G)” paper mentioned here). However, Mahya and I decided to omit mention of this connection, since results in this direction remain partial/incomplete, and since the present paper is long enough.

Further comments and updates might be posted at this link although presently I am short on energy/spare time to write more.

## A minor grumble

(post updated 2021-05-24 with some minor rewording and elaboration)

While I am more sympathetic towards copy-editors than most Very Online Mathematicians, I was a bit unhappy with the final production on the published article (this is not really the fault of individuals, but of the template and workflow). Many of the formulas in the original document were typeset to work at 11 point font size; since the published version appears to be at a larger font size, with font substitution rather than recompiling the LaTeX, these formulas were either shrunk or required line breaks in odd places. Had a suitable style file been provided, we could have taken care of some of this reformatting before the “final version” was sent to the production team.

There is also an issue concerning the order of items in the bibliography, caused by a mismatch between the BibTeX template used in the accepted version and the bibliographic style adopted by the journal. This probably doesn’t bother anyone apart from me, but despite my efforts to get the ordering changed at the page proofs stage, it seems that this instruction was too complicated or obscure for the production team.

I would therefore recommend reading/printing the arXiv version instead, with one caveat: that version has a non-serious but potentially annoying typo, details of which can be found via this page.

tags: ,

Given the state of the world and the plight of many … here it’s a pretty good day, so far.

26. Y. Choi. A gap theorem for the ZL-amenability constant of a finite group. Int. J. Group Th. 5 (2016) no. 4, 27–46.

27. Y. Choi. Realization of compact spaces as cb-Helson sets. Ann. Funct. Anal. 7 (2016), no. 1, 158–169.

28. Y. Choi. Triviality of the generalized Lau product associated to a Banach algebra homomorphism. Bull. Austral. Math. Soc. 94 (2016), no. 2, 286–289.

29. Y. Choi, M. Ghandehari, H. L. Pham. Stability of characters and filters for weighted semilattices. Semigroup Forum 102 (2021), no. 1, 86–103.