Trace-order and a distortion theorem for linearly invariant families on the unit ball of a finite dimensional JB∗ -triple

We give a distortion theorem for linearly invariant families on the unit ball BB of a finite dimensional JB∗-triple XX by using the trace-order. The exponents in the distortion bounds depend on the Bergman metric at 0. Further, we introduce a new definition for the trace-order of a linearly invariant family on BB, based on a Jacobian argument. We also construct an example of a linearly invariant family on BB which has minimum trace-order and is not a subset of the normalized convex mappings of BB for dimX≥2dimX≥2. Finally, we prove a regularity theorem for linearly invariant families on BB. All four types of classical Cartan domains are the open unit balls of JB∗-triples, and the same holds for any finite product of these domains. Thus the unit balls of JB∗-triples are natural generalizations of the unit disc in CC and we have a setting in which a large number of bounded symmetric homogeneous domains may be studied simultaneously.