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Journal of Operator Theory

Volume 74, Issue 1, Summer 2015  pp. 45-74.

$p$-Operator space structure on Feichtinger-Figa-Talamanca-Herz Segal algebras

Authors:  Serap Oztop (1) and Nico Spronk (2)
Author institution:(1) Department of Mathematics, Faculty of Science, Istanbul University, 34134 Vezneciler, Istanbul, Turkey
(2) Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

Summary: We consider the minimal boundedly-translation-invariant Segal algebra $\mathrm{S}_0^p(G)$ in the Fig\{a}-Talamanca-Herz algebra $\mathrm{A}_p(G)$ of a locally compact group $G$. In the case that $p=2$ and $G$ is abelian this is the classical Segal algebra of Feichtinger. Hence we call this the Feichtinger-Fig\{a}-Talamanca-Herz Segal algebra of $G$. This space is also a Segal algebra in $\mathrm{L}^1(G)$ and is, remarkably, the minimal such algebra which is closed under pointwise multiplication by $\mathrm{A}_p(G)$. Even for $p=2$, this result is new for non-abelian $G$. We place a $p$-operator space structure on $\mathrm{S}_0^p(G)$ based on work of Daws (\textsc{M.~Daws}, \textit{J. Operator Theory} \textbf{63}(2010), 47-83), and demonstrate the naturality of this by showing that it satisfies all natural functorial properties: projective tensor products, restriction to subgroups and averaging over normal subgroups. However, due to complications arising within the theory of $p$-operator spaces, we are forced to work with weakly complete quotient maps and weakly complete surjections (a class of maps we define).

Keywords: Figa-Talamanca-Herz algebra, $p$-operator space, Segal algebra

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