It is well known that the open unit ball U of a C*-algebra is a homogeneous (in fact symmetric) space under the action of the Lie group of all biholomorphic automorphisms of U. In finite dimensions, the canonical invariant Riemannian structure is hyperbolic.
In this talk we present results that show what can be said in infinite dimensions. Important here is to use an invariant Hilbert C*-structure on the fibers of the tangent bundle of U. We show that the symmetric spaces 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 same result holds. It also turns out that the invariant connection relates to the Hilbert C*-structure in quite the same way as the Levi-Civita connection does to the Riemannian metric.
These and results joint with D. Blecher can be used to treat invariant cone fields, a structure, which in the realm of general relativity is responsible for the concept of causality.
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Last updated 2 May 2008. This page was created and is maintained by Stephen Wills (s.wills@ucc.ie).