2019/11/11 by Abulhamil, Doha Adel, Jamjoom, Fatmah B., Peralta, Antonio M. · 1 citation
#15A86 (Secondary) #46L05 #46L57 #47B47 (Primary) #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.1911.04134
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 + b T(a)=0); (b) There exist an anti-derivation d:A→ X** and an element ξ∈ X** satisfying ξa = a ξ, ξ[a,b]=0, T(a b) = b T(a) + T(b) a - b ξa, 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.