On maps preserving strongly zero-products

The notion of a strongly zero-product preserving map on normed algebras recently was introduced by the author. This notion is a generalization of the well-known notion "zero-product preserving map." We give a characterization of strongly zero-product preserving maps on normed algebras and also by giving some illustrative and interesting examples. We show that this notion is completely different from the notion of zero-product preserving maps. Also we show that the direct product of two strongly zero-product preserving maps is again a strongly zero-product preserving map. But the tensor product of them need not be a strongly zero-product preserving map. Finally we show that every $*$-preserving linear map from a normed $*$-algebra into a $C*$-algebra that strongly preserves zero-products is necessarily continuous.