2017/06/26 by Ahlem Ben Ali Essaleh, Essaleh, Ahlem Ben Ali, Antonio M. Peralta +1 · 2 citations
Mathematics · Engineering · #Advanced Topics in Algebra #Advanced Operator Algebra Research #Molecular Junctions and Nanostructures
paper · pdf · doi:10.48550/arxiv.1706.08326
We prove new results on generalized derivations on C^*-algebras. By\nconsidering the triple product a,b,c =2-1 (a b^* c + c b^* a), we\nintroduce the study of linear maps which are triple derivations or triple\nhomomorphisms at a point. We prove that a continuous linear T map on a unital\nC^*-algebra is a generalized derivation whenever it is a triple derivation at\nthe unit element. If we additionally assume T(1)=0, then T is a\n^*-derivation and a triple derivation. Furthermore, a continuous linear map\non a unital C^*-algebra which is a triple derivation at the unit element is a\ntriple derivation. Similar conclusions are obtained for continuous linear maps\nwhich are derivations or triple derivations at zero.\n We also give an automatic continuity result, that is, we show that\ngeneralized derivations on a von Neumann algebra and linear maps on a von\nNeumann algebra which are derivations or triple derivations at zero are all\ncontinuous even if not assumed a priori to be so.\n