Invariant Means on $VN^n(G)$

TL;DR

The paper extends invariant-mean theory from $VN(G)$ to the multidimensional setting $VN^n(G)$, establishing non-emptiness of the invariant-mean set $\operatorname{TIM}^n(\widehat{G})$ and revealing a dichotomy: uniqueness for discrete groups and potential non-uniqueness for non-discrete groups. It demonstrates explicit constructions in the discrete case, analyzes how invariant means interact with open subgroups, and connects invariant means on $VN^n(G)$ to those on the cb-multiplier-closure dual $A_0^n(G)$ via canonical maps that yield a bijection. The results provide a robust, functorial picture of invariant means across multidimensional Fourier algebras, their duals, and corresponding closures, with implications for operator-algebraic harmonic analysis on locally compact groups.

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.

Theorems & Definitions (54)

• Definition 3.1
• Definition 3.2
• Lemma 3.3
• proof
• Theorem 3.4
• proof
• Lemma 3.5
• proof
• Proposition 3.6
• proof