On a class of Hilbert C*-manifolds
by
Wend Werner
Westfälische Wilhelms-Universität, Mathematisches Institut, Einsteinstraße 62, 48149 Münster, Germany

Denote by U the open unit ball of a C^*-algebra. U is a symmetric space, where the transitively operating Lie group consists of all biholomorphic automorphisms of U.

This talk has two objectives, an explicit calculation, for all vector fields on U, of the invariant connection and, using results previously obtained with D. Blecher, to characterize those invariant cone fields that can be thought of as the result of some kind of `quantization'. (In general relativity, such a structure is responsible for the concept of causality.)

Both questions are related since the invariance of the cone fields is intimately connected to the behavior of parallel transport along geodesics.

Our results actually cover a much broader class of (infinite dimensional) symmetric spaces. Important here is to use an invariant Hilbert C^*-structure on the fibers of the tangent bundle of U. We show that the symmetric space we are dealing with can be defined in terms of the automorphism group of this structure. For the underlying invariant (operator space) Finsler structure, the analogous result holds. It also turns out that the connection we are dealing with relates to the Hilbert C^*-structure in quite the same way as the Levi-Civita connection does to its Riemannian metric.

Date received: February 27, 2008