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Can’t start a fire without a spark

22 September, 2021

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|


\displaystyle   {\rm AD}(G) - \frac{|\Omega_1|}{|G|}   = \sum_{n\geq 2} n^3 |\Omega_n|   \geq 2\sum_{n\geq 2} n^2 |\Omega_n|  = 2 \left( 1- \frac{|\Omega_1|}{|G|} \right)\qquad(*)

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:

  1. AD(G) = 3/2;
  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.


\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].

For F’s sake

15 September, 2021

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.

Catching up on some belated announcements

23 May, 2021

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.

The Answer to the Ultimate Question

8 March, 2021
tags: ,

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

Mind the gap

11 February, 2021

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.

My role in the band is to be in the middle of that, kind of like lukewarm water

7 January, 2021

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.”

Last orders at the bar

29 December, 2020

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.

Do you regard finite groups as compact or discrete?

11 December, 2020

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…)

It seemed like a clever idea at the time

9 December, 2020

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
  4. Look at what you’ve just produced.
  5. Re-evaluate your life choices.

Abstraction: does removing the clutter also remove the motivation?

7 December, 2020

(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.