Wiener's theorem gives the necessary and sufficient conditions for the set of translates of a set of functions to be dense in $L^1(\mathbb{R}^n)$, which translates algebraically into a statement about the closed ideals of the $C^*$ algebra, $L^1(\mathbb{R}^n)$. It is known that $L^1(G)$ is a $C^*$ algebra for general groups, and that the algebra is commutative iff $G$ is. In the case $G$ is abelian, the result implies that the PNT is equivalent to $\zeta(1+it)\ne 0$ for $t\in\mathbb{R}$ after using Wiener's theorem to prove the Wiener-Ikehara theorem, and ultimately provides a very efficient proof of this result, ultimately depending on the fact that functions with non-vanishing Fourier transforms (on $\mathbb{R}^n$) are those which are not in any maximal ideal of the Fourier algebra.

In the case $G\ne\mathbb{R}^n$ but is still abelian, there is still a notion of "Fourier" transform, more appropriately called the Gelfand transform, which is a genuine generalization of the classical Fourier transform. In the case $G$ is not commutative, then this paper provides a notion of Gelfand transform for the algebra which generalizes the one for the case $G$ is abelian, now with characters of $G$ replaced by unitary representations. If one has, say, a compact group, $G$, which is ** not** abelian, what kinds of analogous results can be obtained from a set of functions $\{f_i\}_{i\in I}\subseteq L^1(G)$ such that their generalized Gelfand transforms have no common zero? What kind of tauberian theorems come out of this, if any?