I find my name has come up in someone’s comment in reaction to an initiative relating to MathOverflow and, while I don’t wish to seem ungracious, the passage of time has robbed me of a lot of the idealism or the energy for standing in between the Montagues and the Capulets.

There is also the dispiriting feeling of being co-opted by a “side” whose values I don’t really share, just because I haven’t jumped on a bandwagon against them.

Tove Jansson summing up how I feel, via Moomintroll.

Update 2021-01-21:

“Good and bad is tricky,” she said. “I ain’t too certain about where people stand. P’raps what matters is which way you face.”

Rounding off 2020 with the third of this year’s solo papers, which has its roots in an idea from 2017 that I couldn’t quite get to work at the time, but where enough partial results emerged this year that it seemed worth writing them up for submission.

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.

That said: this is a preliminary version (not yet submitted), so comments and corrections are welcome.

## 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).

Our approach uses a minorant for AM(A(G)), related to the antidiagonal in G× G, which was implicitly used in Runde’s work but which has not been studied systematically before. 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. We also establish some general properties of this minorant, and present some examples to support the conjecture that the minorant coincides with the amenability constant.

Every locally compact group admits a so-called Haar measure: this is a positive Radon measure on the Borel sigma-algebra of the group, which is invariant under left translations. These conditions determine the Haar measure uniquely up to a choice of positive scaling constant.

Examples of locally compact groups include compact groups and discrete groups.

1. What’s the “natural” normalization of Haar measure for compact groups?
2. What’s the “natural” normalization of Haar measure for discrete groups?

(This post brought to you after a panicked hour yesterday discovering compensating errors in a formula/proof, and a tedious couple of hours today spent rederiving the formulas for Fourier transform and Fourier inversion for $G \times G$ when $G$ is a finite group. It turns out that the identity $1^2=1$ is rather dangerous…)

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Your scientists were so preoccupied with whether or not they could, they didn’t stop to think if they should.

1. Take a nice, infinite compact group such as T2.
2. Regard it as a discrete group T2d, by forgetting the topology.
3. Take the Bohr compactification of this discrete group, obtaining a new compact group
(T2d)^)d)^.
4. Look at what you’ve just produced.

(Completely unrelated to the previous blogpost.)

This post is an experiment, in a way. When teaching mathematics or when writing certain kinds of professional communications (journal articles, but also course notes or reference works) one often seeks to reduce duplication by stating and proving general results. Often these take a form that is much more abstract than the intended applications, and one positive side is that by removing specific features that are irrelevant to the chain of logical reasoning in the proof, one avoids the danger of “not seeing the wood for the trees”.

However, there is the risk that by distilling what one perceives as the key argument into a minimalist and abstract form, one loses both the original context and the motivation for the particular hypotheses chosen.

Of course, there is no right or wrong side here; one needs to allow both perspectives. But I wondered whether the following example, which arises from distilling an argument I cooked up in some recent attempts at research, looks too abstract and artificial, or whether readers might find the hypotheses and objects relate to “natural” examples they encounter in their own research.

In what follows $\mathbb N$ denotes the set of natural numbers, starting from $1$ (apologies to any passing set theorists). As a parallel experiment, I’ve tried to be a little more detailed than I would be as a “working analyst”, so that what follows could be read by students who still wish to see i’s dotted and t’s crossed.

Let $(\mathcal S,\preceq)$ be a partially ordered set (which, for the purposes of what follows, one should think of as uncountable). Suppose $\mathcal S$ has the following properties:

(F) if $A, B\in \mathcal S$ then there exists $D\in \mathcal S$ with $A\preceq D$ and $B\preceq D$;

(σ) if $A_1\preceq A_2 \preceq \dots$ is an increasing sequence in $\mathcal S$, then there exists $C\in \mathcal S$ with $A_n\preceq C$ for all $n\in\mathbb N$.

(Here I am using the analyst’s convention that “increasing” does not mean “strictly increasing”; I am not sure if this is standard usage in the setting of posets.)

Now suppose we have a function $h: {\mathcal S} \to {\mathbb R}$ which is monotone in the sense that $A\preceq B$ implies $h(A)\leq h(B)$, and bounded above in the sense that $\sup_{A\in\mathcal S} h(A) < \infty$.

Claim: $h$ attains its supremum at some element of $\mathcal S$. That is, there exists $C\in {\mathcal S}$ such that $h(C) = \sup_{A\in \mathcal S} h(A)$.

Proof. Let $K=\sup_{A\in\mathcal S} h(A)$. Pick $A_1\in \mathcal S$. We inductively construct $A_2, A_3, \dots$ in $\mathcal S$ which satisfy $A_n \preceq A_{n+1}$ and $h(A_{n+1}) > K- 1/n$ for all $n\in \mathbb N$.

For the inductive step: given $n\in \mathbb N$ and $A_n\in\mathcal S$, pick some $B_n\in \mathcal S$ such that $h(B_n) > K - 1/n$ (possible by the definition of $K$) and then use the hypothesis (F) to obtain $A_{n+1}\in \mathcal S$ satisfying $A_n \preceq A_{n+1}$ and $B_n \preceq A_{n+1}$; monotonicity of $h$ ensures that $h(A_{n+1})\geq h(B_n) > 1- 1/n$.

Having obtained this sequence $(A_n)_{n\in\mathbb N}$, the hypothesis (σ) ensures that there exists $C\in \mathcal S$ such that $A_n\preceq C$ for all $n\in \mathbb N$. Then for each $n\in\mathbb N$ we have $K \geq h(C) \geq h(A_n) > K-1/n$, and we conclude that $h(C)=K$ as required. QED.

Question for readers: if you saw this result in a paper or a book that you were reading, followed by a single application which is much more concrete and specialized, would you feel happy with this? Or would you prefer to see the original argument as it occurred “in the wild”, followed by a remark that there is a more “Platonic” ideal form that could be formulated?

Bonus points, by the way, if you manage to guess the original setting for the argument which gave rise to what’s written above.

It should be fairly clear that I don’t blog here as much as I once intended, and that even when I do there isn’t really enough mathematics. This post is intended as a placeholder or index for any future posts that I might write which relate to the theory of “liquid vector spaces” that emerge from Clausen & Scholze’s framework of “condensed mathematics”.

My interest was sparked by

• (n-category cafe, 2020-03-28) Corfield: Pyknoticity versus cohesiveness (see discussions in comments)
• (Xena project, 2020-12-05) guest post by Scholze: Liquid tensor experiment (which is very lucid and even-handed in its exposition, as well as being interesting in the context of Buzzard’s larger project)

since for a few years during and after my PhD studies, I became interested in what might be the “proper” setting for the homological approach to Banach bimodules over Banach algebras as pioneered by Helemskii and his Moscow school. This interest led me to observe, over the years, a number of people independently realise or propose that Functional Analysis as classically defined is working in “the wrong category”, and to then claim with varying degrees of tongue-in-cheek badinage or missionary zeal that Y’all Only Have Difficulties Because Y’all Insist On Doing Things The Wrong Way.

My perspective at the time is alluded to in this 2007 post. Despite the mild snark above, I am actually by the standards of functional analysts quite partial to “abstract nonsense”, as one might detect from my first published paper; but at heart I still get a kick from inequalities and explicit formulas, and remain a believer that it is worth knowing how to take a long but low-tech approach to given problems.

Tangentially, I still want to see a proof of the known result that the “tensor-flat” Banach spaces are precisely the “script L-1” spaces of Lindenstrauss and Pelczynski, which somehow runs parallel to the proof of the Govorov-Lazard theorem that flat modules are filtered colimits of finitely generated free modules. (There was something along these lines in work of Aristov, IIRC, but I never quite satisfied myself that it was what I was after.) Perhaps this would follow from a suitable embedding of the category of Banach spaces and “short” linear maps into the category of liquid vector spaces and whatever-the-morphisms-are?

Minor update 2020/12/13: upon re-reading Scholze’s post, I couldn’t help doing a double-take when I saw this passing remark:

When combining bounds with homological algebra, it means that within all our complexes we have to carefully keep track of norms. This quickly gets nasty. Did you ever try to chase norms in the snake lemma? In a spectral sequence?

Erm … yes? I mean, perhaps not in such a fancy setting, but see e.g.

9am EDT, November 6th 2020

On a related theme: it’s good to remember from time to time that mathematics can still occasionally be about exploration and not just about fracking.

## A short proof that B(L1) is not amenable

Non-amenability of B(E) has been surprisingly difficult to prove for the classical Banach spaces, but is now known for E= lp and E=Lp for all 1 ≤ p < ∞. However, the arguments are rather indirect: the proof for L1 goes via non-amenability of l(K(l1)) and a transference principle developed by Daws and Runde (Studia Math., 2010). In this note, we provide a short proof that B(L1) and some of its subalgebras are non-amenable, which completely bypasses all of this machinery. Our approach is based on classical properties of the ideal of representable operators on L1, and shows that B(L1) is not even approximately amenable.

Update 2020-11-06: To appear in Proceedings of the Royal Society of Edinburgh, Section A. Published online as DOI: 10.1017/prm.2020.79

tags:

It’s been a while; long enough that I feared I’d forgotten how to write a solo paper. Put another way: