Product of operators and numerical range preserving maps

Let V be the C*-algebra B(H) of bounded linear operators acting on the Hubert space H, or the Jordan algebra S(H) of self-adjoint operators in B(H). For a fixed sequence (i 1 , . . ., i m) with i 1, . . ., i m ∈ {1, . . ., k}, define a product of A 1, . . ., A k∈ V by A 1 * ⋯ * A k = A i1 ⋯ A im. This includes the usual product A 1 * ⋯ * A k = A 1 ⋯ * A k 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 μ m = 1 such that φ: V → V has the form A H → μU* AU or A → μU* A tU, then φ is surjective and satisfies W(A 1 * ⋯ * A k) = W(φ(A 1) * ⋯ * φ (A k)) for all A 1, . . ., A k ∈ V. It is shown that the converse is true under the assumption that one of the terms in (i 1, . . ., i m) is different from all other terms. In the finite-dimensional case, the converse can be proved without the surjectivity assumption on φ. An example is given to show that the assumption on (i 1, . . ., i m) is necessary.