# 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).

**DOI: **http://dx.doi.org/10.7900/jot.2014apr30.2046

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

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