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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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