# Which groups are the unitary group of a $C^*$-algebra

Which groups are the unitary group of a $C^*$-algebra?

Does anyone know anything in this direction?

• Excellent question! I've been wondering for a while, too, but there seems to be very little available. In finite dimension, U(n) is a Lie group and can be characterised as such. Otherwise, about the only literature that I know of is Kadison's "Infinite unitary groups" – Chris Heunen Jan 4 '14 at 23:12

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.

• You might also take the reduced C*-algebra. But in my view it says nothing about the OP (with due respect). – Alain Valette Jan 8 '14 at 21:10
• Well, the idea is that it shows that the class of groups that arises as unitary groups of a C*-algebra cannot be too restricted. Not a full answer, but the best I can think of. – Chris Heunen Jan 13 '14 at 10:48

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.

• But this is a different question, isn't it? – Joël Jan 5 '14 at 22:34
• @Joël --- the objection is that the OP is not asking about group $C^{\ast}$ algebra's? – Carlo Beenakker Jan 7 '14 at 12:55
• No, the objection is that the OP is asking about the range of the map {C*-algebra} --> {topological groups} while your answer discusses the injectivity of that map. – Joël Jan 7 '14 at 14:12