In what topology is the unitary group of a unital C*-algebra locally compact?

Question

If a unital $C^*$-algebra $\mathcal{A}$ is finite-dimensional then the unitary group $\mathcal{U}(\mathcal{A})$ is compact with respect to the norm topology. My question is: what if $\dim(\mathcal{A})=\infty$? Is there a topology on $\mathcal{A}$ for which $\mathcal{U}(\mathcal{A})$ is locally compact? Do I consider $\mathcal{A}$ with the relative $w^*$-topology inherited from $\mathcal{A}^{**}$? Hints would be appreciated.

Answer 1

Since every element $x$ of the closed unit ball $B$ of a $C^*$ algebra can be written as $(1/2) \sum_{j=1}^4 u_j$ where $u_1, \ldots, u_4$ are unitary, if $\mathcal U(\mathcal A)$ is compact (in a topology where $\mathcal A$ is a topological vector space and $B$ is closed) then so is $B$. A $C^*$-algebra $\mathcal A$ is a $W^*$-algebra iff it can be considered as a dual Banach space. In that case you can use the $\sigma$-topology, i.e. the weak-* topology on $\mathcal A$ considered as this dual Banach space. If it's not a $W^*$-algebra, I don't think $\mathcal U(\mathcal A))$ can be compact in any reasonable topology.