On an isomorphism theorem for the Feichtinger's Segal algebra on locally compact groups

TL;DR

This paper shows that the structure of a locally compact group can be fully recovered from the algebraic properties of its Feichtinger's Segal algebra, establishing an isomorphism theorem based on algebraic preservation.

Contribution

It proves that algebraic isomorphisms of Feichtinger's Segal algebras correspond to homeomorphic isomorphisms of the underlying groups, revealing a deep connection between algebraic and topological structures.

Findings

Algebraic properties of $S_0(G)$ determine the group $G$.

Any algebraic bijection preserving products corresponds to a group isomorphism.

The result extends to non-continuous bijections, emphasizing algebraic structure's importance.

Abstract

In this article we observe that a locally compact group $G$ is completely determined by the algebraic properties of its Feichtinger's Segal algebra $S_0(G).$ Let $G$ and $H$ be locally compact groups. Then any linear (not necessarily continuous) bijection of $S_0(G)$ onto $S_0(H)$ which preserves the convolution and pointwise products is essentially a composition with a homeomorphic isomorphism of $H$ onto $G.$