The angular derivative of Fréchet-holomorphic maps in J*-algebras and complex Hilbert spaces

We study the problem of the existence of the angular derivative of Fréchet-holomorphic maps of the generalized right half-planes defined by non-zero partial isometries in infinite dimensional complex Banach spaces called J^*-algebras. These generalized right half-planes and open unit balls (which are bounded symmetric homogeneous domains) are holomorphically equivalent. The results obtained here also hold in C ^*-algebras, JC^*-algebras, B ^*-algebras and ternary algebras, containing non-zero partial isometries, and in complex Hilbert spaces. Some examples are given. The principal tool we use are general results of the Pick-Julia type in J^*-algebras.