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Star order on JBW algebras. (English) Zbl 1371.17027

Summary: The goal of the paper is to extend the star order from associative algebras to non-associative Jordan Banach structures. Let \(\mathcal A\) be a JBW algebra. We define a relation on \(\mathcal A\) as the set of all pairs \((a,b)\in\mathcal A\times\mathcal A\) such that the range projections of \(a\) and \(b-a\) are orthogonal. We show that this relation defines a partial order on \(\mathcal A\) which, in the case of the self-adjoint part of a von Neumann algebra, gives the star order. After showing basic properties of this order we shall prove the following preserver theorem: Let \(\mathcal A\) be a JBW algebra without Type \(\mathrm I_2\) direct summand and let \(\varphi\) be a continuous map from \(\mathcal A\) to \(\mathcal B\) preserving the star order in both directions. If for each scalar \(\lambda\) one has \(\varphi(\lambda 1)=f(\lambda)z\), where \(f\) is a (continuous) function and \(z\) is a central invertible element, then there is a unique Jordan isomorphism \(\psi:\mathcal A\to\mathcal B\) such that \(\varphi(a)=\psi(f(a))z\). Moreover, we show that if \(\mathcal A\) is a Type \(\mathrm I_n\) factor, where \(n\neq 2\), then the equation above holds for all continuous maps preserving the star order in both directions.

MSC:

17C65 Jordan structures on Banach spaces and algebras
47C15 Linear operators in \(C^*\)- or von Neumann algebras
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