# At the third stroke

**Further update 2022-12-06:** after I wrote the first version of this post, I discovered an extra reference that I should have cited in the bibliography, and which would have made my life easier had I known about it before submitting! See the body of the post for further details.

An update: the paper “An explicit minorant for the amenability constant of the Fourier algebra”, which has been mentioned in previous posts on 2020-12-29 and 2021-12-29, has recently been accepted for publication in International Mathematical Resarch Notices IMRN (the redundancy in the name is not a typo).

The “author accepted manuscript”, as we must now call these things in the UK’s Procrustean REF-linked open access framework, can be found on the arXiv as 2012.14413. Since the abstract has been tweaked slightly compared to the submitted ~~previous~~ version, here is the updated abstract:

We show that if a locally compact group G is non-abelian then the amenability constant of its Fourier algebra is ≥ 3/2, extending a result of Johnson (JLMS, 1994) who proved that this holds for finite non-abelian groups. Our 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 the amenability constant, 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.

The accepted version benefited significantly from a detailed referee’s report, which suggested some technical improvements in the discussion of measurability issues, and also gave a sketch of the following result: if A and B are C^{*}-algebras, and D denotes the completion of A ⊗ B in some C^{*}-tensor norm, then the canonical map is a homeomorphism onto its range. (There are known examples, with both A and B beng C^{*}-algebras of discrete groups, where the range of this map is not closed in the codomain.) The application relevant to this paper is as follows: applying this result when A and B are the full group C^{*}-algebras of Type I groups G and H, in conjuction with some standard results, one deduces that the unitary dual of G × H can be naturally identified (as a topological space) with the product of the respective duals. (Somewhat irritatingly, this conclusion is not stated anywhere in Dixmier’s C^{*}-book, even though I suspect it must have been known to people working on group C^{*}-algebras at the time the book was written.)

**Update 2022-12-06:** it turns out that the C^{*}-algebraic result referred to above can be found in one of the appendices to Raeburn and Williams’s book on Morita Equivalence and Continuous-Trace C^{*}-Algebras, which is not a book I own and not a source I would have thought to turn to. The arguments they use are essentially the same as the one I came up with in the appendix of my paper based on the referee’s sketch. However, since they develop various things from scratch to keep their account more self-contained, I decided to keep the appendix to my paper (which outsources a lot of standard details to Dixmier’s book) and merely add a note to acknowledge the Raeburn-Williams book.

In an earlier version of the paper (and in the 2021-12-29 blogpost) I mentioned the following question: *if G is a finite group whose amenability constant is less than or equal to 2, do all its irreps have degree at most 2?* It turned out, in between the submission and the acceptance, that the answer is yes, but I decided to leave that for a future paper (perhaps for future blog posts).