Geometric properties of systems of vector states and expansion of states in Pettis integrals (Q2811874)

The setting of the paper is formed by a separable Hilbert space \(H\), the algebra \(B(H)\) of all bounded linear operators on \(H\), its state space together with the well-known unique decomposition of a state into its normal and singular part, the latter also being called the purely finitely additive part. Parallelly, there are the space \(ba(\mathbb{N})\) (identified with the dual of \(\ell^\infty\)) of complex-valued finitely additive measures on \(\mathbb{N}\) (considered with the \(\sigma\)-algebra of all subsets of \(\mathbb{N}\)), its state space \(W(\mathbb{N})\) (of all positive finitely additive positive measures of norm 1) and the possibility to decompose each element of \(ba(\mathbb{N})\) uniquely as the sum of its countably additive and its purely finitely additive part.NEWLINENEWLINEIn this context the authors cite from previous papers that for any (in particular non-normal) state \(\omega\) on \(B(H)\) there are a countable family \(E=\{e_n\mid n\in\mathbb{N}\}\) of unit vectors of \(H\) and a measure \(\mu\in W(\mathbb{N})\) such that NEWLINE\[NEWLINE\omega=\int_\mathbb{N} \rho_{e_n}\;d\mu(n)\tag{1}NEWLINE\]NEWLINE where the integral is meant as \(\omega(x)=\int_\mathbb{N} \rho_{e_n}(x)\;d\mu(n)\) for all \(x\in B(H)\).NEWLINENEWLINEThe paper addresses aspects of the interdependence of properties of \(E\), \(\omega\) and \(\mu\).NEWLINENEWLINEFor example it is shown that if \(\omega\) in (1) is positive purely finitely additive then so is \(\mu\), and if \(\mu\) is countably additive then \(\omega\) is normal; if the \(e_n\) converge weakly to \(0\) then also the converses of both implications hold.NEWLINENEWLINEThe ``pure states'' of \(ba(\mathbb{N})\) (i.e. the extremal points of \(W(\mathbb{N})\)) are exactly the \(\{0;1\}\)-valued measures in \(W(\mathbb{N})\). The authors show that if \(\omega\) is a pure state on \(B(H)\) then it admits an expansion (1) where \(\mu\) is \(\{0;1\}\)-valued. Given a \(\{0;1\}\)-valued measure in \(W(\mathbb{N})\), the authors give sufficient conditions on \(E\) for \(\omega\) to have a non vanishing normal part or to have a non vanishing singular part.NEWLINENEWLINEThe countable family \(\mathcal{S}_E\) of vector states associated to \(E\) is called pseudominimal if \(\mu\) in (1) is unique. In this case a criterion for extremality of \(\mu\) in \(W(\mathbb{N})\) is given. Moreover, if \(E\) is, for example, an orthogonal basis of \(H\) then \(\mathcal{S}_E\) is pseudominimal.NEWLINENEWLINEReviewer's remark: The existence of (1) is cited from Theorem 2 of [\textit{V. Zh. Sakbaev}, Russ. Math. 55, No. 10, 41--50 (2011); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2011, No. 10, 48--58 (2011; Zbl 1229.81078)] where in the proof it is claimed that the operator norm and the numerical radius of a bounded linear operator are equal; this is true for selfadjoint operators whereas in general the two norms are only equivalent. But it seems that (1) is not really affected as it can be recovered from showing it first for selfadjoint \(x\). (Similarly for Theorem 1 of [\textit{G. G. Amosov} and \textit{V. Zh. Sakbaev}, Math. Notes 93, No. 3, 351--359 (2013); translation from Mat. Zametki 93, No. 3, 323--332 (2013; Zbl 1269.81008)].)