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Probability measures in -algebras in Hilbert spaces with conjugation

Authors:	Marjan Matvejchuk

Institution:	Department of Mechanics and Mathematics, Kazan State University, 18 Lenin St., 420008, Kazan, Russia

Abstract:	Let $\mathcal{M}$ be a real $W^{*}$ -algebra of -real bounded operators containing no central summand of type $I_{2}$ in a complex Hilbert space with conjugation . Denote by the quantum logic of all -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 $\mu (p)=\tau (p)$ , $\forall p\in P$ .

Keywords:	Quantum logics measure Hilbert space $W^*$-algebra

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