A NOTE ON LOCAL COMMUTATORS IN DIVISION RINGS WITH INVOLUTION

In this paper, we consider a conjecture of I. N. Herstein for local commutators of symmetric elements and unitary elements of division rings. For example, we show that if D is a finite dimensional division ring with involution * and if a is an element of D* = D\{0} such that local commutators axa(-1) x(-1) at a are radical over the center F of D for every x is an element of D* with x* = x, then either a is an element of F or dim(F) D = 4.