Isometric isomorphism of the annihilator of \(C_0(G)\) in \(LUC(G)^*\)

Abstract

Let \(LUC(G)\) denote the \(C^*\)-algebra of left uniformly continuous functions with the uniform norm and let \(C_0(G)^{\perp}\) denote the annihilator of \(C_0(G)\) in \(LUC(G)^*\). 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.