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On a class of Hilbert C*-manifolds 2234.4. In the following we will suppose that ‘space’ is represented by a certain set D ofself-adjoint operators, and that on this set there is defined an invariant cone field. Thequestion we would like to answer is this: How can such fields be characterized, whichcome from interpreting the points of D as bounded Hilbert space operators?In the following, denote by E a fixed (abstract) ternary ring of operators, and by Uits unit ball.4.5. On top of the causal structure of U we need ‘selfadjointness’, which for us will bethe existence of a ‘real form’ for U , compatible with the (almost) complex structure.We do this by requiring that E carries an involutory real automorphism ‘*’ so that.ix/ D ix for all x 2 E. Since we will be studying TRO-embeddings into L.H /that preserve the real form, it will be necessary to impose the additional condition that,for all x;y;z 2 E, fx;y;zg Dfz ;y ;x g. (Note that then each x 2 E has a uniquedecomposition into real and imaginary parts, with some norm estimates.) A ternaryring of operators E that meets all these conditions, will be called a *-ternary ring ofoperators. We will suppose in the following that E is a space of this kind.4.6. The ‘space manifold’here will be the open unit ball U sa of the selfadjoint part of E.U sa is itself a symmetric space. If G sa consists of all elements in Aut U which leaveU sa invariant, then U sa D G sa =H sa , where H sa comprises the TRO-automorphisms thatare *-selfadjoint. In fact, it follows from fx;y;zg Dfz ;y ;x g for all x;y;z 2 E(and an expansion into power series) that M a .x/ D M a .x / for all x 2 U and soM a 2 G sa iff a D a. A TRO-automorphism T is in H sa iff T.x / D T.x/ for allx 2 U , and soG sa D fT ı M a j T 2 H sa ;a 2 U sa g ;as well as U sa D G sa =H sa .4.7. In order to comply with the requirement that causality be invariant under paralleltransport we have to impose the condition that the field of cones we fix in TU sa mustbe invariant under the action of G sa . We consider smooth embeddings ˆW U ! L.H /which preserve• the Hilbert C*-structure,• the complex structure as well as the (canonical) real forms,• the action of the automorphism groups.And we want to know: What characterizes the G sa invariant cone fields that are pulledback to U via ˆ? Whenever a cone field meets these properties we will call it natural.4.8. Since a natural cone field is supposed to be invariant under the action of G sa ,we may restrict our attention to cones in T o U D E. Furthermore, any cone in Ethat gives rise to a G sa invariant field of cones has to be invariant under the action ofH sa . It can also be shown that under the assumptions made above, dˆ has to preservethe ternary structure of each tangent space T p U . The question we were asking thusbecomes: What properties must an H sa -invariant cone in E sa possess so that it is of the