Revista Matemática Iberoamericana

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Volume 34, Issue 4, 2018, pp. 1469–1514
DOI: 10.4171/rmi/1033

Published online: 2018-12-03

$p$-Fourier algebras on compact groups

Hun Hee Lee[1], Ebrahim Samei[2] and Nico Spronk[3]

(1) Seoul National University, Republic of Korea
(2) University of Saskatchewan, Saskatoon, Canada
(3) University of Waterloo, Canada

Let $G$ be a compact group. For $1\leq p\leq\infty$ we introduce a class of Banach function algebras $\mathcal A^p(G)$ on $G$ which are the Fourier algebras in the case $p=1$, and for $p=2$ are certain algebras discovered by Forrest, Samei and Spronk. In the case $p\neq 2$ we find that $\mathcal A^p(G)\cong\mathcal A^p(H)$ if and only if $G$ and $H$ are isomorphic compact groups. These algebras admit natural operator space structures, and also weighted versions, which we call $p$-Beurling–Fourier algebras. We study various amenability and operator amenability properties, Arens regularity and representability as operator algebras. For a connected Lie $G$ and $p > 1$, our techniques of estimation of when certain $p$-Beurling–Fourier algebras are operator algebras rely more on the fine structure of $G$, than in the case $p=1$. We also study restrictions to subgroups. In the case that $G=$ SU(2), restrict to a torus and obtain some exotic algebras of Laurent series. We study amenability properties of these new algebras, as well.

Keywords: Compact group, Fourier series, operator weak amenability, operator amenability, Arens regularity

Lee Hun Hee, Samei Ebrahim, Spronk Nico: $p$-Fourier algebras on compact groups. Rev. Mat. Iberoam. 34 (2018), 1469-1514. doi: 10.4171/rmi/1033