Linear Maps Which are Anti-derivable at Zero

Abstract

Let T : A → X be a bounded linear operator, where A is a C *-algebra, and X denotes an essential Banach A-bimodule. We prove that the following statements are equivalent: (a) T is anti-derivable at zero (i.e., ab = 0 in A implies T (b)a + bT (a) = 0); (b) There exist an anti-derivation d : A → X * * and an element ξ ∈ X * * satisfying ξ a = aξ, ξ [a, b] = 0, T (ab) = bT (a) + T (b)a − bξ a, and T (a) = d(a) + ξ a, for all a, b ∈ A. We also prove a similar equivalence when X is replaced with A * *. This provides a complete characterization of those bounded linear maps from A into X or into A * * which are anti-derivable at zero. We also present a complete characterization of those continuous linear operators which are *-anti-derivable at zero.