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Gleason’s theorem and its applications. (English) Zbl 0795.46045
Mathematics and Its Applications. East European Series. 60. Dordrecht: Kluwer Academic Publishers. Bratislava: Ister Science Press. xv, 325 p. (1993).
Let \(H\) be a Hilbert space, let \(L(H)\) be the lattice of all closed subspaces of \(H\). A mapping \(m: L(H)\to [0,1]\) such that \(m(H)=1\) and \(m(\bigoplus_{i=1}^ \infty M_ i)= \sum_{i=1}^ \infty m(M_ i)\) (here by \(\bigoplus_{i=1}^ \infty M_ i\) we mean the join of a family of mutually orthogonal subspaces \(\{M_ n\}_{n=1}^ \infty\) of \(L(H)\)) is called a state on \(L(H)\). In 1957 A. M. Gleason published the solution of Mackey’s problem about the description of the set of all states on \(L(H)\). He showed, that for every state \(m\) on \(L(H)\) (\(H\) is separable and \(\dim H\neq 2\)) there exists a unique positive Hermitian trace operator \(T\) on \(H\) with \(\text{tr}(T)=1\), such that \(m(M)= \text{tr}(TP_ M)\), \(M\in L(H)\), where \(P_ M\) denotes the orthoprojector from \(H\) onto \(M\) and \(\text{tr}(T)\) is the trace of \(T\). The present book is devoted to the detailed exposition of the proof of Gleason’s theorem and its different generalizations and applications.
The book consists of five chapters. In the first chapter the elements of Hilbert space theory and linear operator theory in Hilbert spaces are described. The choice of facts here is defined by the needs of Gleason’s theorem and its generalizations and applications.
The elements of the quantum logic theory are given in the second chapter.
Gleason’s theorem is proved in the third chapter where also some variants of this theorem for infinite measures and signed measures are established.
In Chapter 4 different measure-theoretic completeness criteria of inner product spaces (using Gleason’s theorem) are proved. For example, it is proved that the existence of at least one non-zero completely additive signed measure implies the completeness.
In Chapter 5 applications of Gleason’s theorem to the investigation of the properties of orthogonal vector-valued measures and their connection to morphisms on quantum logics are given. In the end of the chapter states on the set of all splitting subspaces of so-called Keller spaces (a special kind of the quadratic spaces) are described.
Throughout the book some sections conclude with exercises that illustrate the previous facts or notions and also present problems.
It should be pointed out that the choice of material given in the book is defined by the interests of the author. Unfortunately, there are not enough descriptions of the non-trivial variants of Gleason’s theorem for von Neumann algebras and weakly closed Jordan algebras.

46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
03G12 Quantum logic
46G10 Vector-valued measures and integration