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 ξa=aξ, ξ[a,b]=0, T(ab)=bT(a)+T(b)a−bξa, and T(a)=d(a)+ξa, 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.