Linear Maps Which are Anti-derivable at Zero

Doha Adel Abulhamil,Fatmah B Jamjoom,Antonio M Peralta

doi:10.1007/s40840-020-00918-7

Doha Adel Abulhamil, Fatmah B Jamjoom + Show 1 more

Open Access

https://doi.org/10.1007/s40840-020-00918-7

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Abstract

Let T:Arightarrow X be a bounded linear operator, where A is a hbox {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:Arightarrow X^{**} and an element xi in X^{**} satisfying xi a = a xi ,xi [a,b]=0,T(a b) = b T(a) + T(b) a - b xi a, and T(a) = d(a) + xi a, for all a,bin 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.

Highlights

Let us begin this note by formulating a typical problem in recent studies about preservers
A derivation D is called inner if there exists x0 ∈ X such that D(a) = δx0 (a) = [a, x0] = ax0 − x0 a for all a ∈ A
A typical challenge on preservers can be posed in the following terms: Problem 1 Suppose T : A → X is a linear map satisfying (1.1) only on a proper subset D ⊂ A2