Isometric isomorphism of the annihilator of $C_0(G)$ in $LUC(G{)}^{\ast }$

Let $LUC(G)$ denote the $C^{\ast }$-algebra of left uniformly continuous functions with the uniform norm and let $C_0(G{)}^{\perp }$ denote the annihilator of $C_0(G)$ in $LUC(G{)}^{\ast }$. In this article, among other results, we show that if $G$ is a locally compact group and $H$ is a discrete group then whenever there exists a weak-star continuous isometric isomorphism between $C_0(G{)}^{\perp }$ and $C_0(H{)}^{\perp }$, $G$ is isomorphic to $H$ as a topological group. In particular, when $H$ is discrete $C_0(H{)}^{\perp }$ determines $H$ within the class of locally compact topological groups.