N.-Sing, Product of operators and numerical range preserving maps

The authors dedicate this paper to Professor Miroslav Fiedler on the occasion of his 80th birthday. Let V be the C∗-algebra B(H) of bounded linear operators acting on the Hilbert space H, or the Jordan algebra S(H) of self-adjoint operators in B(H). For a fixed sequence (i1,..., im) with i1,..., im ∈ {1,..., k}, define a product of A1,..., Ak ∈ V by A1 ∗ · · · ∗ Ak = Ai1... Aim. This includes the usual product A1 ∗ · · · ∗ Ak = A1 · · ·Ak and the Jordan triple product A ∗ B = ABA as special cases. Denote the numerical range of A ∈ V by W (A) = {(Ax, x) : x ∈ H, (x, x) = 1}. If there is a unitary operator U and a scalar µ satisfying µm = 1 such that φ: V → V has the form A 7 → µU∗AU or A 7 → µU∗AtU, then φ is surjective and satisfies W (A1 ∗ · · · ∗ Ak) =W (φ(A1) ∗ · · · ∗ φ(Ak)) for all A1,..., Ak ∈ V. It is shown that the converse is true under the assumption that one of the terms in (i1,..., im) is different from all other terms. In the finite dimen-sional case, the converse can be proved without the surjective assumption on φ. An example is given to show that the assumption on (i1,..., im) is necessary