In quantum measure theory, a measure on the projection lattice of a von Neumann algebra – and the corresponding closed subspaces of an Hilbert space – is assumed to be additive with respect to mutually orthogonal subspaces. The states of a quantum physical system, on the other hand, are associated to probability measures on this projection lattice, and a natural example of a probability measure on projections is the restriction of a normal positive linear functional on the von Neumann algebra \(B(H)\) of the bounded operators acting on a Hilbert space \(H\). Whether or not, all probability measures are of this very form was one of the basic questions of mathematical foundations of quantum mechanics, and in this context the celebrated A. M. Gleason Theorem [J. Math. Mech. 6, 885–893 (1957; Zbl 0078.28803)] states that every bounded completely additive complex measure on the lattice of the projections acting on a Hilbert space \(H\) of dimension no lesser than 3 extends to a normal functional on the von Neumann algebra \(B(H)\). Since any normal functional on a von Neumann algebra can be represented by a trace class operator, the Gleason Theorem gives a characterization of any bounded completely additive measure on the projection lattice by means of a trace class operator: this version of the theorem is well known to the physicists, and the trace class operator is usually called the density matrix of the physical state.
The Gleason Theorem has many physically relevant consequences. One of its most famous applications is the proof of nonexistence of dispersion-free state on the Hilbert space logic, and consequently that a Hilbert space model cannot be completed to a classical model by considering suitable auxiliary hidden variables without violating its inner structure. A completely additive measure, however, may be unbounded and hence not extendable into a bounded linear functional; it is remarkable then – in view of the stated relevance of the Gleason type theorems – that complete additivity and boundedness may differ only in a finite dimensional case: S. Dorofeev [J. Funct. Anal. 103, No. 1, 209–216 (1992; Zbl 0759.46056)] and A. N. Sherstnev [Methods of bilinear forms in non-commutative measure and integral theory (Russian). Moskva: Fizmatlit (2008; Zbl 1198.46002)] proved indeed that any completely additive measure on projections in an infinite-dimensional Hilbert space is automatically bounded: this indeed allows one to relax the assumption on the boundedness of measure in the Gleason Theorem and opens the way to extend it. As a consequence we can say that any completely additive measure on the projection lattice of the algebra \(B(H)\) of all bounded operators on an infinite-dimensional Hilbert space \(H\) extends uniquely to a normal functional on \(B(H)\).
In the paper under review – which is completely in the mainstream of the previously outlined research field – the authors first of all establish a Jordan version of Dorofeev’s boundedness result (Theorem 3.1; the long Section III is essentially devoted to its extensive and elaborated proof) stating the boundedness of every completely additive (complex) measure \(\Delta : {\mathcal P}(H(M, \beta))\to {\mathbb C}\) defined on the projection lattice of a suitable \(JBW^\ast\)-algebra \(H(M, \beta)\) which happens to be a Jordan \(^\ast\)-subalgebra of a von Neumann algebra \(M\), while \(\beta:M\to M\) is a \(\mathbb C\)-linear \(^\ast\)-involution. It is instead still an admittedly open problem whether this result remains valid when \(H(M, \beta)\) is replaced by an arbitrary \(JBW^\ast\)-algebra. The second main result (Theorem 4.7) then pertains to the triple derivations on \(JBW^\ast\)-triples, and, in particular, it states that a 2-local triple derivation on a continuous \(JBW^\ast\)-triple always is a linear and continuous triple derivation.
The paper is eminently technical and looks intended mainly for the sector pundits; that notwithstanding, the authors make a commendable effort to preliminarily define in the Section I all the notions and notations that they will use in what follows so that the interested people can undoubtedly delve into their arguments.