Invariant Means on $VN^n(G)$

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.