When working through the proof of Bott-Periodicity (the original proof by Bott), I noticed that compactness of $U(n)$ is important as it gives us that every geodesic looks like an exponential map (something I'm also not quite clear on, and may post as a separate question). Now, I believe that if we are considering the group of unitary matrices with entries in a general $C^{*}$-algebra, we will not always have compactness (this is probably obvious for infinite dimensional algebras, but perhaps for finite dimensional ones we still have compactness?).
I'm not sure whether periodicity will hold in this general case, but an idea that my professor has is to try to show that geodesics will also be exponential maps in this case. He suggested that by studying short segments of geodesics (the geodesic going from some element $a$ to some element $b$), we could perhaps replace them with equally short exponential paths. The idea is that if we can do this for every segment, the new path could be replaced by a shorter path which is exponential from point $a$ to $b$.
I'm not sure how to approach this idea reasonably. Perhaps somebody already has some insight/information regarding periodicity of a more general unitary group, and what geodesics on it look like?