Probability measures in $W^{*}J$-algebras in Hilbert spaces with conjugation

Abstract:

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