Journal of Operator Theory

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

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

Authors: Serap Öztop (1), 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: $ \def\bl{\mathrm{L}} \def\apg{\mathrm{A}_p(G)} \def\sopg{\mathrm{S}_0^p(G)} $ We consider the minimal boundedly-translation-invariant Segal algebra $\sopg$ in the Figà-Talamanca-Herz algebra $\apg$ 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-Figà-Talamanca-Herz Segal algebra of $G$. This space is also a Segal algebra in $\bl^1(G)$ and is, remarkably, the minimal such algebra which is closed under pointwise multiplication by $\apg$. Even for $p=2$, this result is new for non-abelian $G$. We place a $p$-operator space structure on $\sopg$ based on work of Daws M. Daws, J. Operator Theory 63$($2010$)$, 4783 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).

DOI: http://dx.doi.org/10.7900/jot.2014apr30.2046
Keywords: Figà-Talamanca-Herz algebra, $p$-operator space, Segal algebra

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