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218 W. Werner2.2. The objects defined above coincide with the so called ternary rings of operators(TRO), which are intrinsically characterized in [9], [11]. On such a space E, a tripleproduct f; ; g is given in such a way that E, up to (the obvious definition of) TROisomorphisms,is a subspace of a space of bounded Hilbert space operators L.H /,invariant under the triple productfx;y;zg Dxy z:The relation to (left) Hilbert C*-modules is based on the equationfx;y;zg Dhx;yiz;connecting triple product to module action as well as scalar product of a HilbertA-module. Here, A is the algebra of linear mappingsA D EE D lin fx 7! fe 1 ;e 2 ;xgje 1;2 2 Eg ;which is independent of the chosen embedding. Note that the norm of an element e 2 Emust coincide with kek Dkfe; e; egk 1=3 .2.3. TRO-morphisms will be those mappings that preserve the product f; ; g. Thesemappings differ in general from what is considered to be the natural choice for HilbertA-morphisms, the so called adjointable maps. The latter are in particular A-modulemorphisms, a property that would be too restrictive for our purposes. We will hencestick in the following to TRO-morphisms.Definition 2.4. Let M be a Banach manifold and A a C*-algebra. M is said to bea(right-, left-) Hilbert A-manifold if on each tangent space T p .M / there is given thestructure of a Hilbert A-module depending smoothly on base points.Definition 2.5. Let M be a Hilbert C*-manifold. The group of automorphisms,Aut M consists of all diffeomorphisms ˆW M ! M so that dˆ is (pointwise) aTRO-morphism.2.6. It is not clear under which circumstances a Banach manifold can be given thestructure of a Hilbert C*-manifold. This is so because, first, no characterization seemsto be known of Banach spaces that are (topologically linear) isomorphic to HilbertC*-modules, and, second, because in general, there is no smooth partition of the unitthat would permit the step from local to global.We will use group actions instead. This will bring up homogeneous Hilbert C*-manifolds in the following sense.Definition. A Banach manifold M is called homogeneous Hilbert C*-manifold iff it isa Hilbert C*-manifold for which Aut M acts transitively.2.7. Suppose M is a homogeneous space with respect to a smooth Banach Lie groupaction. Fix a base point o 2 M , and denote the isotropy subgroup at o by H . Supposethat T o .M / carries the structure of a Hilbert A-module with form h; i o , module