Résume | I will present a way how to study (connected) Banach-Lie groups from the point of view of large scale geometry. The original idea comes from C*-algebra theory; namely, we got inspired by Ringrose'notion of `C*-exponential length' for unitary groups of C*-algebras, thoroughly studied in the previous decades by Lin, Phillips, and others. It turns out that the exponential length well-defines a quasi-isometry type of any connected Banach-Lie group (and actually it has good both large scale and small scale properties). After discussing some generalities, I will focus on Banach-Lie examples of groups with properties (T), (FH) and of Haagerup. For the latter, we believe these should be the first non-trivial and non-abelian ones among non-locally compact groups. I will also present some perhaps counter-intuitive non-examples: while for n>2 and any unital commutative Banach algebra A, SL(n,A) always has property (T), SL(2,A) doesn't have the Haagerup property if A is an infinite-dimensional separable unital commutative C*-algebra. It is joint work with Hiroshi Ando and Yasumichi Matsuzawa. |