C∗-Isomorphisms, Jordan Isomorphisms, and Numerical Range Preserving Maps

Let V = B(H) or S(H), where B(H) is the algebra of bounded linear operator acting on the Hilbert space H, and S(H) is the set of self-adjoint operators in B(H). Denote the numerical range of A ∈ B(H) by W (A) = {(Ax, x) : x ∈ H, (x, x) = 1}. It is shown that a surjective map φ : V→ V satisfies W (AB +BA) =W (φ(A)φ(B) + φ(B)φ(A)) for all A,B ∈ V if and only if there is a unitary operator U ∈ B(H) such that φ has the form X 7→ ±U∗XU or X 7→ ±U∗X U, where X t is the transpose of X with respect to a fixed orthonormal basis. In other words, the map φ or −φ is a C∗-isomorphism on B(H) and a Jordan isomorphism on S(H). Moreover, if H has finite dimension, then the surjective assumption on φ can be removed. 2000 Mathematics Subject Classification. 47A12, 47B15, 47B49, 15A60, 15A04, 15A18