A Characterization of Intrinsic Functions on

Abstract

Let be an associative algebra over the field and let be the group of all automorphisms and anti-automorphisms of which leave elementwise invariant. A function Fwith domain and range contained in is called an intrinsic functionon if (i) for each Ω in and (ii) F(ΩZ) = ΩF(Z) for every Z in .Rinehart (5) has introduced and motivated the study of the class of intrinsic functions on , and has characterized these functions for the cases in which is the algebra of real quaternions, the algebra of n × ncomplex matrices, or the algebra of n× nreal matrices (5; 6). The algebras listed above, along with the algebra of n × nquaternion matrices, constitute the full list of possibilities for the simple direct summands of any semi-simple algebra over or ; see (2).