$p$-Operator space structure on Feichtinger--Fig\`{a}-Talamanca--Herz Segal algebras

TL;DR

This paper introduces a $p$-operator space structure on a class of Segal algebras associated with locally compact groups, extending classical results to non-abelian groups and demonstrating natural functorial properties.

Contribution

It defines a $p$-operator space structure on the Feichtinger--Fig ext`a-Talamanca--Herz Segal algebra and proves its naturality and functorial properties, including for non-abelian groups.

Findings

The space is a minimal Segal algebra in $A_p(G)$ and $L^1(G)$.

The $p$-operator space structure satisfies functorial properties.

Results hold even for non-abelian groups, with weakly completely bounded maps.

Abstract

We consider the minimal boundedly-translation-invariant Segal algebra $S_0^p(G)$ in the Fig\`{a}-Talamanca--Herz algebra $A_p(G)$ 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 $G$. Remarkably, this space is also a Segal algebra in $L^1(G)$ and is, in fact, 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 $S_0^p(G)$, and demonstrate the naturality of this by showing that it satisfies all natural functiorial 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 completely bounded maps in many of our results.