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Disjointness preservers of \(AW^\ast\)-algebras. (English) Zbl 1398.46046
Summary: In this paper, we study bijective linear maps \(\theta : A \rightarrow B\) between \(A W^\ast\)-algebras preserving either zero products or range orthogonality. Such a map is automatically continuous, and provides an algebra or a \(\ast\)-algebra isomorphism \(\pi(\cdot) = \theta(\cdot) \theta^{\ast \ast}(1)^{- 1}\) from \(A\) onto \(B\). Our results extend previous works for the case of \(W^\ast\)-algebras, and also cover such maps between \(C^\ast\)-algebras satisfying an extra assumption on preserving abelian \(C^\ast\)-subalgebras.

MSC:
46L10 General theory of von Neumann algebras
47B48 Linear operators on Banach algebras
46L40 Automorphisms of selfadjoint operator algebras
46H40 Automatic continuity
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