$M_d$-multipliers of a locally compact group

TL;DR

This paper characterizes the space of $M_d$-multipliers of a locally compact group as a dual space of a specific tensor product, revealing structural insights and inheritance properties of approximation properties.

Contribution

It provides an isometric isomorphism between $M_d(G)$ and a dual space of a tensor product, advancing understanding of $M_d$-multipliers and their approximation properties.

Findings

$M_d(G)$ is isometrically isomorphic to the dual of a tensor product space.

$M_d$-type-approximation-properties are inherited to lattices.

Structural characterization of $M_d$-multipliers in terms of Banach space duals.

Abstract

We show that the space $M_d(G)$ of $M_d$-multipliers of a locally compact group $G$ is isometrically isomorphic to the Banach space of bounded functionals on the $d$-fold Haagerup tensor product of $L^1(G)$ vanishing on the kernel of the convolution map. Consequently, we see that $M_d(G)$ is isometrically isomorphic to the dual space of $X_d(G)$, the completion of $L^1(G)$ in the dual of $M_d(G)$. We also show that $M_d$-type-approximation-properties are inherited to lattices.