On Boolean algebras of projections and prespectral operators (Q1820413)

Let \(B\) be a Boolean algebra of projections \(E\) in a complex Banach space \(X\). Let G be a total linear manifold in the dual space \(X^*\). Let \(B*G=span(T*g\); \(T\in B^ c\), \(g\in G)\), where \(B^ c\) is the commutant of \(B\) in the algebra \(B(X)\) of all bounded linear operators in \(X\). \(B\) is called \(G\)-complete if (1) \(B\) is complete as an abstract Boolean algebra; (2) for every set \(B_ 0\) in B, (\(\bigvee (E\); \(E\in B_ 0))X\) is the closure of the linear hull of \((EX; E\in B_ 0)\) in the weak topology \(w(X,B*G)\), and \((\bigwedge(E;E\in B_ 0))X = \cap(EX;E\in B_ 0);\) (3) each monotonic net \(B_ 0\) in b becomes small on small sets with respect to the \(w(X,B*G)\)-topology. A similar definition is given for \(G\) to be called \(G\)-\(\sigma\)-complete. The author gives a characterization of \(G\)-complete (\(G\)-\(\sigma\)- complete) Boolean algebras of projections in terms of monotonic nets (sequences). He states that a Boolean algebra of projections is \(G\)- \(\sigma\)-complete if and only if it coincides with the range of a spectral measure of class \((M,G)\) for some \(\sigma\)-algebra \(M\) of subsets of a compact Hausdorff space. Let \(T\in B(X)\). Some conditions, which involve the resolution of the identity (spectral measure) for \(T^*\) or \(T^{**}\), are given for \(T\) to be prespectral or spectral. The results of the paper are an extension of the work of \textit{W. G. Bade}: On complete Boolean algebras of projections [Trans. Am. Math. Soc. 80, 345-360 (1955; Zbl 0066.362)].