Large scale geometry of Banach-Lie groups

Abstract

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-localy compact groups having the Haagerup property, most of them being non-amenable. These are the groups U2(M, τ ), where M is a semifinite von Neumann algebra with a normal faithful semifinite trace τ . Finally, we investigate the groups En(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 E2(A) does not have the Haagerup property. If A is moreover abelian and separable, then SL(2, A) does not have the Haagerup property.

On epimorphisms in some categories of infinite-dimensional Lie groups

“…We show that not (2) ⇒ not (1). Suppose f : G → K is a morphism such that some b ∈ K commutes with f (H) but not with f (G).…”