Elementary operators on self-adjoint operators ✩

Let H be a Hilbert space and let A and B be standard ∗-operator algebras on H. Denote by As and Bs the set of all self-adjoint operators in A and B, respectively. Assume that M:As →Bs and M∗ :Bs →As are surjective maps such that M(AM∗(B)A) = M(A)BM(A) and M∗(BM(A)B) = M∗(B)AM∗(B) for every pair A ∈ As , B ∈ Bs . Then there exist an invertible bounded linear or conjugate-linear operator T :H →H and a constant c ∈ {−1, 1} such that M(A) = cT AT ∗, A ∈ As, and M∗(B) = cT ∗BT , B ∈ Bs . © 2006 Elsevier Inc. All rights reserved.