Final published version
Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - An explicit minorant for the amenability constant of the Fourier algebra
AU - Choi, Yemon
PY - 2023/11/30
Y1 - 2023/11/30
N2 - We show that if a locally compact group $G$ is non-abelian, then the amenability constant of its Fourier algebra is $\geq 3/2$, extending a result of [9] 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 [16]. To do this, we study a minorant for the amenability constant, related to the anti-diagonal in $G\times G$, 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.
AB - We show that if a locally compact group $G$ is non-abelian, then the amenability constant of its Fourier algebra is $\geq 3/2$, extending a result of [9] 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 [16]. To do this, we study a minorant for the amenability constant, related to the anti-diagonal in $G\times G$, 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.
KW - Amenability constant
KW - Fourier algebra
KW - Group theory
U2 - 10.1093/imrn/rnac348
DO - 10.1093/imrn/rnac348
M3 - Journal article
VL - 2023
SP - 19390
EP - 19430
JO - International Mathematics Research Notices
JF - International Mathematics Research Notices
SN - 1073-7928
IS - 22
ER -