ON FUNCTIONAL IDENTITIES IN PRIME-RINGS WITH INVOLUTION

Let A be a prime ring with involution , let S be the symmetric elemen ts, let K be the skew elements, let Q(ml) be the maximal left ring of quotients, x(1),..., x(m) noncommuting variables, and E-i, F-j, G(k), H-l: A(m-1) --> Q(ml), i, j, k, l = 1,2,..., m. We study functional id entities of the form Sigma(i=1)(m) E(i)(i)x(i) + Sigma(j=1)(m)x(j)F(j) (j) + Sigma(k=1)(m)G(k)(k)x(k)() + Sigma(l=1)(m)x(l)(*)H(l)(l) = 0 fo r all x(1),..., x(m) is an element of A (where E-i(i) means E-i(x(1),. .., (x) over cap(i),..., x(m)), etc.). In case S boolean OR K is not a lgebraic of bounded degree less than or equal to 2m definitive results are obtained. As an application k-commuting traces of symmetric n-add itive maps of either S or K into Q(ml) are characterized. (C) 1998 Aca demic Press.