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Mappings preserving zero products. (English) Zbl 1032.46063
Let $$A$$ and $$B$$ be associative algebras with unit element over a field and $$\theta:A\to B$$ a linear map. Then $$\theta$$ is said to be a zero-product preserving map is $$\theta(a)\theta(b)= 0$$ whenever $$ab= 0$$ in $$A$$. For example, if $$h$$ is in the center of $$A$$ and $$\phi: A\to B$$ is an algebra homomorphism, then $$\theta= h\phi$$ is zero-product preserving. In the present paper, it is shown that in many interesting cases zero-product preserving linear maps arise in this way. Applications of these results are given to matrix algebras, standard operator algebras, $$C^*$$-algebras and $$W^*$$-algebras.

##### MSC:
 46H70 Nonassociative topological algebras 46L40 Automorphisms of selfadjoint operator algebras 47B48 Linear operators on Banach algebras
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