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

TL;DR

This paper investigates the structure of the annihilator of $C_0(G)$ in $LUC(G)^*$ and shows that for certain groups, an isometric isomorphism of these annihilators implies the groups are topologically isomorphic.

Contribution

It establishes that for locally compact groups and discrete groups, a weak-star continuous isometric isomorphism of the annihilators implies the groups are topologically isomorphic.

Findings

Isometric isomorphism of annihilators implies group isomorphism for certain classes.

$C_0(H)^{ot}$ uniquely determines the discrete group $H$.

Results connect algebraic isomorphisms with topological group structures.

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.