Dye's theorem and Gleason's theorem for \(AW^\ast\)-algebras
A famous \(C^*\)-algebraist, in his own defence, once remarked that every \(C^*\)-algebraist should be allowed to write one, but only one, paper on \(AW^\ast\)-algebras. Although the paper under review is an example of the genre, it nevertheless provides an interesting investigation into the properties of \(AW^\ast\)-algebras in which there has been a resurgence of interest in recent years.NEWLINENEWLINE In this paper, the author answers in the positive a question asked by \textit{C. Heunen} and \textit{M. L. Reyes} [J. Math. Anal. Appl. 416, No. 1, 289--313 (2014; Zbl 1310.46053)] of whether a function \(\phi\) from the complete lattice \({\mathcal P}(A)\) of projections in the \(AW^\ast\)-algebra \(A\), that has no type I\(_2\) direct summand, to \({\mathcal P}(B)\), where \(B\) is also an \(AW^\ast\)-algebra, that preserves arbitrary suprema and orthocomplements, automatically satisfies the condition that, for each pair \(p\) and \(q\) of elements in \({\mathcal P}(A)\), NEWLINE\[NEWLINE\phi((e_A- 2p) q(e_A- 2p))= (e_B- 2\phi(p)) \phi(q)(e_B- 2\phi(p)),NEWLINE\]NEWLINE where \(e_A\) and \(e_B\) are the units in \(A\) and \(B\), respectively. It then follows that \(\phi\) extends to a normal Jordan \(^*\)-homomorphism from \(A\) to \(B\). As a consequence of this result, he is able to generalize Dye's theorem to \(AW^\ast\)-algebras and show that the Mackey-Gleason-Bunce-Wright theorem also holds for a large class of \(AW^\ast\)-algebras. He then goes on to connect these results with his own work on order isomorphisms between the partially ordered sets of abelian sub-\(C^*\)-algebras of unital \(C^*\)-algebras.
scientific article; zbMATH DE number 6362866
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