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.

]]>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.”

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.

]]>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.

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

- What’s the “natural” normalization of Haar measure for compact groups?
- 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 when is a finite group. It turns out that the identity is rather dangerous…)

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

- Take a nice, infinite compact group such as
**T**^{2.} - Regard it as a discrete group
**T**^{2}_{d}, by forgetting the topology. - Take the Bohr compactification of this discrete group, obtaining a new compact group

(**T**^{2}_{d})^)_{d})^. - Look at what you’ve just produced.
- Re-evaluate your life choices.

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 denotes the set of natural numbers, starting from (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 be a partially ordered set (which, for the purposes of what follows, one should think of as uncountable). Suppose has the following properties:

(F) if then there exists with and ;

(σ) if is an increasing sequence in , then there exists with for all .

(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 which is monotone in the sense that implies , and bounded above in the sense that .

**Claim:** attains its supremum at some element of . That is, there exists such that .

*Proof.* Let . Pick . We inductively construct in which satisfy and for all .

For the inductive step: given and , pick some such that (possible by the definition of ) and then use the hypothesis (F) to obtain satisfying and ; monotonicity of ensures that .

Having obtained this sequence , the hypothesis (σ) ensures that there exists such that for all . Then for each we have , and we conclude that 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.

]]>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

]]>Non-amenability of B(E) has been surprisingly difficult to prove for the classical Banach spaces, but is now known for E= l

_{p}and E=L_{p}for all 1 ≤ p < ∞. However, the arguments are rather indirect: the proof for L_{1}goes via non-amenability of l^{∞}(K(l_{1})) and a transference principle developed by Daws and Runde (Studia Math., 2010). In this note, we provide a short proof that B(L_{1}) 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 L_{1}, and shows that B(L_{1}) 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