Probability measures in 𝑊*𝐽-algebras in Hilbert spaces with conjugation

Let M \mathcal {M} be a real W ∗ W^{*} -algebra of J J -real bounded operators containing no central summand of type I 2 I_{2} in a complex Hilbert space H H with conjugation J J . Denote by P P the quantum logic of all J J -orthogonal projections in the von Neumann algebra N = M + i M {\mathcal {N}}={\mathcal {M}}+ i{\mathcal {M}} . Let μ : P → [ 0 , 1 ] \mu :P\rightarrow [0,1] be a probability measure. It is shown that N \mathcal {N} contains a finite central summand and there exists a normal finite trace τ \tau on N \mathcal {N} such that μ ( p ) = τ ( p ) \mu (p)=\tau (p) , ∀ p ∈ P \forall p\in P .