On a theorem of Brauer-Cartan-Hua type

We shall be concerned here with the nature of subrings of a ring with involution which are invariant with respect to certain combinations of elements. To be more precise, let R be a ring with involution * and suppose that A is a subring of R such that xAx* c A for all x 6 R. Can we say something definitive about the structure of A We shall see that if R is semi-prime then we do get a dichotomy of the Brauer-Cartan-Hua type, namely, A must contain a nonzero ideal of R or A must be central.