A theorem about states on quantum logics

On Two Problems Concerning Enveloping von Neumann Algebras of JW-Algebras

“…Therefore extension of a measure, defined on projections of the JW-algebra with no type I 2 direct summand to a state, defined on this JW-algebra is equivalent to extension of a measure, defined on projections of this JW-algebra to a measure defined on projections of the enveloping von Neumann algebra of this JW-algebra. On the problem of extension of a measure to a linear functional on Jordan operator algebras have been written a series of works (for example, [2][3][4][5][6][7]). The second problem considering in the given work is interesting and actual up to now.…”

Skew-symmetric functions on the hyperboloid and quantum measures

“…His profound work, which was fundamental for all subsequent advances in this area, considered positive, countably additive quantum measure on B(H), where H is a separable Hilbert space and dim H --> 3. The solution for a yon Neumann algebra of type III or II= and for a positive quantum measure was first given by the conjunction of the work of Christensen (1982) and the one for countably additive positive measures for semifinite von Neumann algebras (Matvejchuk, 1980). Later, this result was repeated with a similar proof (Yeadon, 1993).…”