On spectral synthesis on element-wise compact Abelian groups

Let $G$ be an arbitrary locally compact Abelian group and let $C(G)$ be the space of all continuous complex-valued functions on $G$. A closed linear subspace $\mathscr H\subseteq C(G)$ is referred to as an invariant subspace if it is invariant with respect to the shifts $\tau_y\colon f(x)\mapsto f(xy)$, $y\in G$. By definition, an invariant subspace $\mathscr H\subseteq C(G)$ admits strict spectral synthesis if $\mathscr H$ coincides with the closure in $C(G)$ of the linear span of all characters of $G$ belonging to $\mathscr H$. We say that strict spectral synthesis holds in the space $C(G)$ on $G$ if every invariant subspace $\mathscr H\subseteq C(G)$ admits strict spectral synthesis. An element $x$ of a topological group $G$ is said to be compact if $x$ is contained in some compact subgroup of $G$. A group $G$ is said to be element-wise compact if all elements of $G$ are compact. The main result of the paper is the proof of the fact that strict spectral synthesis holds in $C(G)$ for a locally compact Abelian group $G$ if and only if $G$ is element-wise compact.
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