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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.

46H70Nonassociative topological algebras
46L40Automorphisms of $C^*$-algebras
47B48Operators on Banach algebras
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