Normal weights on JBW-algebras. (Russian) Zbl 0604.46066
The main result of the paper consists in showing the following theorem being an analogue of the classical result for weights on von Neumann algebras.
Theorem. Let \(\phi\) be a weight on a JBW-algebra A. The following conditions are equivalent:
(i) \(\phi\) is completely additive;
(ii) \(\phi\) is normal;
(iii) \(\phi\) is weakly continuous from below;
(iv) \(\phi (a)=\sup \{\psi (a):\psi \in F\}\), \(a\in A^+\), where F is a family of positive normal functionals on A;
(v) \(\phi\) is a sum of positive normal functionals on A.
Theorem. Let \(\phi\) be a weight on a JBW-algebra A. The following conditions are equivalent:
(i) \(\phi\) is completely additive;
(ii) \(\phi\) is normal;
(iii) \(\phi\) is weakly continuous from below;
(iv) \(\phi (a)=\sup \{\psi (a):\psi \in F\}\), \(a\in A^+\), where F is a family of positive normal functionals on A;
(v) \(\phi\) is a sum of positive normal functionals on A.
Reviewer: A.Luczak
MSC:
46L51 | Noncommutative measure and integration |
46L53 | Noncommutative probability and statistics |
46L54 | Free probability and free operator algebras |
46L70 | Nonassociative selfadjoint operator algebras |
28C15 | Set functions and measures on topological spaces (regularity of measures, etc.) |