Large scale geometry of Banach-Lie groups

We initiate the large scale geometric study of Banach-Lie groups, especially linear groups. show that exponential length, originally introduced by Ringrose for unitary groups $C^*$-algebras, defines quasi-isometry type any connected group. As an illustrative example, we consider separable abelian unital $C^*$-algebras with spectrum having finitely many components, which classify up to topological isomorphism and quasi-isometry, in order highlight difference. The main results then concern Haagerup property, Properties (T) (FH). present first non-trivial non-abelian non-locally compact most them being non-amenable. These are $\mathcal {U}_2(M,\tau )$, where $M$ is a semifinite von Neumann algebra normal faithful trace $\tau$. Finally, investigate $\operatorname {E}_n(A)$, closed subgroups {GL}(n,A)$ generated elementary matrices, $A$ Banach algebra. $n\geq 3$, all these have Property they unbounded, so (FH) non-trivially. On other hand, if infinite-dimensional $C^*$-algebra, {E}_2(A)$ does not property. If moreover separable, {SL}(2,A)$