Distinguishing C*-algebras by their unitary groups

TL;DR

This paper investigates whether the isomorphism of unitary groups of C*-algebras, considered as topological or discrete groups, can determine the isomorphism class of the C*-algebras themselves.

Contribution

It provides partial affirmative results on whether unitary group isomorphisms imply C*-algebra isomorphisms, advancing understanding of their structural relationship.

Findings

Unitary group isomorphism can imply C*-algebra isomorphism in certain cases.

Partial results support the conjecture in specific contexts.

The study distinguishes between topological and discrete group isomorphisms.

Abstract

We obtain partial affirmative answers to the question whether isomorphism of the unitary groups of two C*-algebras, either as topological groups or as discrete groups, implies isomorphism of the C*-algebras as real C*-algebras.