Large scale geometry of banach-lie groups

We initiate the large scale geometric study of Banach-Lie groups, especially of linear Banach-Lie groups. We show that the exponential length, originally introduced by Ringrose for unitary groups of C*-algebras, defines the quasi-isometry type of any connected Banach-Lie group. As an illustrative example, we consider unitary groups of separable abelian unital C*-algebras with spectrum having finitely many components, which we classify up to topological isomorphism and up to quasi-isometry, in order to highlight the difference. The main results then concern the Haagerup property, and Properties (T) and (FH). We present the first non-trivial non-abelian and non-locally compact groups having the Haagerup property, most of them being non-amenable. These are the groups u(2)(M, tau), where M is a semifinite von Neumann algebra with a normal faithful semifinite trace tau. Finally, we investigate the groups E-n(A), which are closed subgroups of GL(n, A) generated by elementary matrices, where A is a unital Banach algebra. We show that for n >= 3, all these groups have Property (T) and they are unbounded, so they have Property (FH) non-trivially. On the other hand, if A is an infinite-dimensional unital C*-algebra, then E-2(A) does not have the Haagerup property. If A is moreover abelian and separable, then SL(2, A) does not have the Haagerup property.