Invariant Means on $VN^n(G)$

TL;DR

This paper introduces invariant means on the dual of the multidimensional Fourier algebra for locally compact groups, establishing their existence, properties, and invariance under subgroup relations.

Contribution

It defines invariant means on $VN^n(G)$, proves their non-emptiness, and explores their behavior for different types of groups and subgroups.

Findings

Invariant means exist on $VN^n(G)$ for all locally compact groups.

The number of invariant means is preserved under open subgroups.

Invariant means on the dual of $A_0^n(G)$ are also studied.

Abstract

Let $G$ be a locally compact group, and $VN^n(G)$ is the dual of the multidimensional Fourier algebra $A^n(G)$. In this article, we define invariant means on $VN^n(G)$ and prove that the set of all invariant means on $VN^n(G)$ is non-empty. Further, we investigated the invariant means on $VN^n(G)$ for discrete and non-discrete cases of $G$. Also, we show that if $H$ is an open subgroup of $G$, then the number of invariant means on $VN^n(H)$ is the same as that of $VN^n(G)$. Finally, we study invariant means on the dual of the algebra $A_0^n(G)$, the closure of Fourier algebra $A^n(G)$ in the cb-multiplier norm.