Noncommutative Figa-Talamanca-Herz algebras for Schur multipliers

TL;DR

This paper develops a noncommutative analogue of a classical algebra related to Schur multipliers, characterizing isometric multipliers and the structure of bounded Schur multipliers on Schatten spaces.

Contribution

It introduces a noncommutative version of the Figà-Talamanca-Herz algebra for Schur multipliers and characterizes the isometric and bounded multipliers in this setting.

Findings

Identifies the isometric Schur multipliers.

Shows that the space of bounded Schur multipliers is the weak operator topology closure of isometric multipliers.

Establishes a noncommutative analogue of classical harmonic analysis structures.

Abstract

We introduce a noncommutative analogue of the Fig\'a-Talamanca-Herz algebra $A_p(G)$ on the natural predual of the operator space $\frak{M}_{p,cb}$ of completely bounded Schur multipliers on Schatten space $S_p$. We determine the isometric Schur multipliers and prove that the space $\frak{M}_{p}$ of bounded Schur multipliers on Schatten space $S_p$ is the closure in the weak operator topology of the span of isometric multipliers.