Let A be a JBW-algebra (Jordan Banach algebra with the predual), P the complete orthomodular lattice of projections (idempotents) in A. A probability measure on P is a map \(\mu\) : \(P\to [0,\infty)\) with \(\mu (\mathbf{1})=1\) and \(\mu\) (\(\sum_{i}e_ i)=\sum_{i}\mu (e_ i)\) for any orthogonal family \(\{e_ i\}\) in P. The main result of the paper is: Theorem. Let A be a JBW-algebra without type \(I_ 2\) direct summands. Then any probability measure on P can be uniquely extended to a normal state on A.
Remark. A similar result was obtained independently by L. J. Bunce and J. D. Maitland Wright [Commun. Math. Phys. 98, 187-202 (1985; Zbl 0579.46049)].