process

On a class of Hilbert C*-manifolds 219map x 7! a o x, and TRO-structure fx;y;zg o Dhx;yi o o z. If for all h 2 H ,x;y;z 2 T o .M / we haved o hfx;y;zg o Dfd o h.x/; d o h.y/; d o h.z/g othen a Hilbert A-module on T p .M / for any p D g.o/ 2 M , is well-defined throughfx;y;zg p D d o gfd p g 1 .x/; d p g 1 .y/; d p g 1 .z/g o :With this structure, M becomes a homogeneous Hilbert C*-manifold2.8. Our example here is the following. Fix a TRO E and denote by U its open unitball. If we follow the path laid out above, we find the following invariant HilbertC*-structure on U . Define a triple product for T a M at a 2 U byfx;y;zg a D x.1 a a/ 1 y .1 aa / 1 z;so thatas well ashx;yi a D .1 aa / 1=2 x.1 a a/ 1 y .1 aa / 1=2 a z D .1 aa / 1=2 .1 aa / 1=2 ; 2 EE :We will refer to this structure as the canonical Hilbert C*-structure on U .2.9. This definition is motivated in the following way. As shown in [4], Hol U , thegroup of all biholomorphic automorphisms U , consists of mappings of the form T ıM a ,where for any a 2 U ,M a .x/ WD .1 aa / 1=2 .x C a/.1 C a x/ 1 .1 a a/ 1=2 ;and T is a (linear) isometry of E, restricted to U . Then Hol U acts transitively on U ,and the group of (linear) isometries is the isotropy subgroup at the point 0. For lateruse, we include here the fact thatas well asd x M a .h/ D .1 aa / 1=2 .1 C xa / 1 h.1 C a x/ 1 .1 a a/ 1=2dx 2 M a.h 1 ;h 2 /D .1 aa / 1=2 .1 C xa / 1 h 2 a .1 C xa /h 1 .1 C a x/ 1 .1 a a/ 1=2.1 aa / 1=2 .1 C xa / 1 h 1 .1 C a x/ 1 a h 2 .1 C a x/ 1 .1 a a/ 1=2 :The Hilbert C*-structure from 2.8 is in fact constructed according to the constructionin 2.6 for a group G, somewhat smaller than Hol U .