On maps preserving zeros of the polynomial xy − yx*

Abstract Let A  =  M n ( F ) be the matrix algebra over a field F with an involution ∗, where n  ⩾ 20. Suppose that θ  :  A  →  A is a bijective linear map such that θ ( x ) θ ( y ) =  θ ( y ) θ ( x )* for all x ,  y  ∈  A such that xy  =  yx *. We show that θ is of the form θ ( x ) =  λuxu −1 for x  ∈  A , where λ is a nonzero symmetric scalar and u is a normal matrix such that uu * is a nonzero scalar.