Viewing the unit ball in a complex Hilbert space as a set of relativistically admissible complex velocities, the holomorphic automorphisms of the ball emerge as ‘‘complex relativistic velocity additions’’ and ‘‘complex rotations’’ (that is, unitary transformations). Hence, newly discovered properties of the abstract real Lorentz group [A. A. Ungar, Am. J. Phys. 59, 824 (1991); A. A. Ungar, Am. J. Phys. 60, 815 (1992)], which turn out to provide powerful tools for the study of the holomorphic automorphisms of the unit ball in any complex Hilbert space, are of interest in the literature [W. Rudin, Function Theory in the Unit Ball of Cn (Springer-Verlag, New York, 1980) ]. A particularly useful and important tool turns out to be provided by Thomas gyration, which is an abstraction of the well-known Thomas precession of special relativity theory, and which gives rise to a gyrogroup structure underlying the ball. Results of this article set the stage for the study, in a subsequent article, of complex Lorentz transformation groups, U(1,n), in a way analogous to the study of Galilean groups. The dimension n is finite or infinite, and a real parameter c is involved in such a way that in the limit of large c, c→∞, the Lorentz groups that we study reduce to their Galilean counterparts.