Linearity of a charge on a lattice of orthoprojectors
Let \({\mathcal A}\) be a \(JW\)- or \(AW^*\)-algebra and let \(\Pi\) denote the lattice of all (selfadjoint) projections from \({\mathcal A}\). We say that \(\nu:\Pi\to\mathbb{C}\) is a finitely additive charge if \(\nu\) is bounded and finite-additive on mutually orthogonal projections. The author solves the classical Gleason-Mackey problem, i.e. he shows that a finitely additive charge can be extended to a linear norm continuous functional on the whole algebra if \({\mathcal A}\) is one of the following: (i) a \(JW\)-algebra without a direct summand of type \(I_2\); (ii) a purely infinite \(AW^*\)-algebra; (iii) a finite \(AW^*\)-algebra admitting a faithful normal centre-valued trace. For a discussion of the problem, see [\textit{L. J. Bunce} and \textit{J. D. M. Wright}, Bull. Am. Math. Soc., New Ser. 26, No. 2, 288-293 (1992; Zbl 0759.46054)]. This important paper would benefit a lot from a better translation into English.
\(JW\)-algebra without a direct summand of type \(I_ 2\)
scientific article; zbMATH DE number 923036