The second dual of strongly zero-product preserving maps

The notion of strongly Lie zero-product preserving maps on normed algebras as a generalization of Lie zero-product preserving maps are defined. We give a necessary and sufficient condition from which a linear map between normed algebras to be strongly Lie zero-product preserving. Also some hereditary properties of strongly Lie zero-product preserving maps are presented. Finally the second dual of a strongly zero-product, strongly Jordan zero-product and strongly Lie zero-product preserving map on a certain class of normed algebras are investigated.