Linear Maps Which are Anti-derivable at Zero
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C∗ -algebraBanach bimoduleDerivationAnti-derivationMaps ∗ -anti-derivable at zero
bulhamil, D.A., Jamjoom, F.B. & Peralta, A.M. Linear Maps Which are Anti-derivable at Zero. Bull. Malays. Math. Sci. Soc. 43, 4315–4334 (2020). [https://doi.org/10.1007/s40840-020-00918-7]
SponsorshipSpanish Ministry of Science, Innovation and Universities (MICINN); European Regional Development Fund Project PGC2018-093332-B-I00; Junta de Andalucia FQM375; Proyecto de I + D + i del Programa Operativo FEDER Andalucia 2014-2020 A-FQM-242-UGR18; Deanship of Scientific Research (DSR), King Abdulaziz University; DSR
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.