Strong regularity and generalized inverses in jordan systems

A notion of generalized inverse extending that of Moore-Penrose inverse for continuous linear operators between Hilbert spaces and that of group inverse for elements of an associative algebra is defined in any Jordan triple system (J, P). An element a∈J has a (unique) generalized inverse if and only if it is strongly regular, i.e., a∈P(a) 2 J. A Jordan triple system J is strongly regular if and only if it is von Neumann regular and has no nonzero nilpotent elements. Generalized inverses have properties similar to those of the invertible elements in unital Jordan algebras. With a suitable notion of strong associativity, for a strongly regular element a∈J with generalized inverse b the subtriple generated by {a, b} is strongly associative