Which groups are the unitary group of a $C^*$-algebra?
Does anyone know anything in this direction?
Which groups are the unitary group of a $C^*$-algebra?
Does anyone know anything in this direction?
Here is a partial answer: for any given group $G$ there exists a C*-algebra $A$ whose unitary group contains $G$ as a subgroup.
Namely, take $A$ to be the (full) group C*-algebra $C^*(G)$ of $G$. This follows from the fact that the functor $U \colon \mathbf{Cstar} \to \mathbf{Grp}$ taking unitary groups is right adjoint to the functor $C^* \colon \mathbf{Grp} \to \mathbf{Cstar}$ taking (full) group C*-algebras. Moreover, both functors are faithful and reflect isomorphisms. See Nassopoulos.
Characterizing group $C^{\ast}$-algebras through their unitary groups: the Abelian case, J. Galindo and A.M. Rodenas (2007).
We study to what extent group $C^{\ast}$-algebras are characterized by their unitary groups. A complete characterization of which Abelian group $C^{\ast}$-algebras have isomorphic unitary groups is obtained.