On maps preserving zeros of the polynomial xy − yx*

Let A = Mn(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.