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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.

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