I will present joint work with Hiroshi Ando and Yasumichi Matsuzawa on the geometry of Banach-Lie groups. We show that each connected Banach-Lie group admits a distance defining its Lipschitz structure, whose definition originally comes from the C*-algebra theory.
We exhibit several new examples of groups with property (T), resp. the Haagerup property, within this class of groups. We shall also have a closer look at the geometry of unitary groups of C*-algebras and I will mention a new ongoing project with H. Ando on the Lie theoretic investigation of such groups.